Homotopy Type and Homology Homotopy Type and Homology

OXFORD MATHEMATICAL MONOGRAPHS

Series Editors J. M. BALL E. M. FRIEDLANDERI. G. MACDONALD L. NIRENBERG R. PENROSEJ. T. STUART OXFORD MATHEMATICAL MONOGRAPHS

A. Belleni-Moranti: Applied semigroups and evolution equations J.A. Goldstein: Semigroups of linear operators M. Rosenblum and J. Rovnyak: Hardy classes and operator theory J.W.P. Hirschfeld: Finite projective spaces of three dimensions K. Iwasawa: Local class field theory A. Pressley and G. Segal: Loop groups J.C. Lennox and S.E. Stonehewer: Subnormal subgroups of groups D.E. Edmonds and W.D. Evans: Spectral theory and differential operators Wang Jianhua: The theory of games S. Omatu and J.H. Seinfeld: Distributed parameter systems: theory and applications D. Holt and W. Plesken: Perfect groups J. Hilgert, K.H. Hofmann, and J.D. Lawson: Lie groups, convex cones, and semigroups S. Dineen: The Schwarz lemma B. Dwork: Generalized hypetgeometric functions R.J. Baston and M.G. Eastwood: The Penrose transform: its interaction with representation theory S.K. Donaldson and P.B. Kronheimer: The geometry of four-manifolds T. Petrie and J. Randall: Connections, definite forms, and four-manifolds R. Henstock: The general theory of integration D.W. Robinson: Elliptic operators and Lie groups A.G. Werschulz: The computational complexity of differential and integral equations J.B. Griffiths: Colliding plane waves in general relativity P.N. Hoffman and J.F. Humphreys: Projective representations of the symmetric groups 1. Gyorgi and G. Ladas: The oscillation theory of delay differential equations J. Heinonen, T. Kilpelainen, and O. Moartio: Non-linear potential theory B. Amberg, S. Franciosi, and F. de Giovanni: Product of groups M.E. Gurtin: Thennomechanics of evolving phase boundaries in the plane 1. Ionescu and M. Sofonea: Functional and numerical methods in viscoplasticity N. Woodhouse: Geometric quantization 2nd edition U. Grenander: General pattern theory Jacques Faraut and Adam Koranyi: Analysis on symmetric cones I.G. Macdonald: Symmetric functions and Hall polynomials 2nd edition B.L.R. Shawyer and B.B. Watson: Borel's methods of summability M. Holschneider: Wavelets: an analysis tool Jacques Thevenaz: G-algebras and modular representation theory Hans-Joachim Baues: Homotopy type and homology Homotopy Type and Homology

HANS-JOACHIM BAUES

Max-Planck-Institut ftir Mathematik Bonn, Germany

CLARENDON PRESS OXFORD 1996 Oxford University Press, Walton Street, Oxford OX2 6DP Oxford New York Athens Auckland Bangkok Bombay Calcutta Cape Town Dares Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto and associated companies in Berlin Ibadan

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PREFACE

The main problem and the hard core of algebraic topology is the classification of homotopy types of polyhedra. Here the general idea of classification is to attach to each polyhedron invariants, which may be numbers, or objects endowed with algebraic structures (such as groups, rings, modules, etc.) in such a way that homotopy equivalent polyhedra have the same invariants (up to isomorphism in the case of algebraic structures). Such invariants are called homotopy invariants. The ideal would be to have an algebraic invariant which actually characterizes a homotopy type completely. This book represents a new attempt to classify simply connected homotopy types in terms of homology and homotopy groups and additional algebraic structure on these groups. The main new result and our principal objective is the classification theorem in Chapter 3 on `k-invariants' and `boundary invariants' which supplements considerably the classical picture of homology and homotopy groups in the literature. The second part of the book (Chapters 6-12) displays a number of explicit computations of homotopy types which are obtained by applying the classification theorem. In particular J.H.C. Whitehead's classical theorem on 1- connected 4-dimensional homotopy types follows immediately. The old challenging problem of extending Whitehead's classification for 1-connected 5-dimensional homotopy types is solved in Chapter 12. We also classify 2- connected 6-dimensional homotopy typesand (n - 1)-connected (n + 3)- dimensional homotopy types, n >- 4, which are in the stable range (so that the classification does not depend on the choice of n). A complete list of all finite stable (n - 1)-connected (n + 3)-dimensional homotopy types is described in Chapter 10. For example, there are exactly 24 simply connected homotopy types X with homology groups H4(X)=Z/6, H5(X)=71/2, H6(X)=71/2, H7(X)=71 and Hi(X) = 0 otherwise. For n >- 2 we also compute the homotopy types Y with at most three non-trivial homotopy groups zr,,,7r + ,and 7r+ 2. For example, there exist exactly seven homotopy types Y with

ar4(Y) = 71/6, 75(Y) = 71/2, ir6(Y) = 71/2 and iri(Y) = 0 otherwise. We point out that our results for the first time provide methods to compute such homotopy types with three non-trivial homology groups or homotopy groups. viii PREFACE

Such classification results involve computations of low-dimensional homotopy groups, Chapter 11. For this the results on homotopy groups of mapping cones in the Appendix are needed. We obtain complete information on the fourth homotopy group vr4(X) of a simply connected space X and more generally on the homotopy group iri+2 of an (n - 1)-connected space, n > 2. This continues the classical programme of Hurewicz, resp. J.H.C. Whitehead, who achieved such results for the homotopy groups vra, resp. of an (n - 1)-connected space, n >- 2. The book is essentially self-contained; prerequisites are elementary topology and elementary algebra and some basic notions of category theory. It can be used as an introduction to the subject and as a basis of further research. Moreover it provides methods and examples of explicit homotopy classification for those who would like to use such results in other fields, for example for the classification of manifolds. We refer the reader to the survey article (Baues [HT]) for a general outline of the theory of homotopy types in the literature. I would like to acknowledge the support of the Institute for Advanced Study in Princeton. I am also grateful to the staff of Oxford University Press and to the typesetter for their helpful cooperation during the production of this book.

Bonn H.-J.13. June 1995 CONTENTS

Introduction 1

Chapter 1Linear extension and Moore spaces 8 1.1 Detecting functors, linear extensions, and the cohomology of categories 8 1.2Whitehead's quadratic functor F 15 1.3 Moore spaces and homotopy groups with coefficients 18 1.4 Suspended pseudo-projective planes 21 1.5The homotopy category of Moore spaces M", n >_ 3 23 1.6 Moore spaces and the category G 25

Chapter 2Invariants of homotopy types 31 2.1 The Hurewicz homomorphism and Whitehead's certain exact sequence 32 2.2 T-groups with coefficients 40 2.3 An exact sequence for the Hurewicz homomorphism with coefficients 45 2.4Infinite symmetric products and Kan loop groups 51 2.5Postnikov invariants of a homotopy type 54 2.6Boundary invariants of a homotopy type 63 2.7Homotopy decomposition and homology decomposition 72 2.8Unitary invariants of a homotopy type 76

Chapter 3 On the classification of homotopy types 81 3.1 kype functors 81 3.2bype functors 85 3.3Duality of bype and kype 89 3.4The classification theorem 97 3.5 The semitrivial case of the classification theorem and Whitehead's classification 104 3.6The split case of the classification theorem 107 3.7Proof of the classification theorem 111

Chapter 4 The CW-tower of categories 116 4.1 Exact sequences for functors 117 4.2Homotopy systems of order (n + 1) 121 X CONTENTS

4.3The CW-tower of categories 124 4.4Boundary invariants for homotopy systems 129 4.5Three formulas for the obstruction operator 131 4.6 A-Realizability 135 4.7Proof of the boundary classification theorem 138 4.8The computation of isotropy groups in the CW-tower 143

Chapter 5Spanier-Whitehead duality and the stable CW-tower 149 5.1 Cohomotopy groups 149 5.2Spanier-Whitehead duality 152 5.3Cohomology operations and homotopy groups 156 5.4The stable CW-tower and its dual 163

Chapter 6Eilenberg-Mac Lane and Moore functors 168 A Eilenberg-Mac Lane functors 168 6.1 Homology of Eilenberg-Mac Lane spaces 168 6.2Some functors for abelian groups 170 6.3Examples of Eilenberg-Mac Lane functors 178 6.4On (m - 1)-connected (n + 1)-dimensional homotopy types with ir,X=Oform

B Moore functors 190 6.7Moore types and Moore functors 191 6.8On (m - 1)-connected (n + 1)-dimensional homotopy types X with H; X = 0 for m < i < n 195 6.9The stable case with trivial 2-torsion 196 6.10Moore spaces and Spanier-Whitehead duality 198 6.11 Homotopy groups of Moore spaces in the stable range 202 6.12 Stable and principal maps between Moore spaces 206 6.13 Quadratic Z-modules 215 6.14 Quadratic derived functors 225 6.15 Metastable homotopy groups of Moore spaces 229

Chapter 7The homotopy category of (n -1)-connected (n + 1)-types 239 7.1 A linear extension for types;, 240 7.2The enriched category of Moore spaces 244 CONTENTS xi Chapter 8 On the homotopy classification of (n -1)-connected (n + 3) -dimensional polyhedra, n >_ 4 249 8.1 Algebraic models of (n - 1)-connected (n + 3)-dimensional homotopy types, n > 4 249 8.2On M(A, n) 258 8.3The group F.12of an (n - 1)-connected space, n >- 4 263 8.4Proof of the classification theorem 8.1.6 267 8.5Adem operations 269

Chapter 9 On the homotopy classification of 2-connected 6-dimensional polyhedra 277 9.1Algebraic models of 2-connected 6-dimensional homotopy types 277 9.2On 7r5M(A,3) 286 9.3Whitehead's group I5 of a 2-connected space 291

Chapter 10Decomposition of homotopy types 294 10.1 The decomposition problem in representation theory and topology 294 10.2 The indecomposable (n - 1)-connected (n + 3)-dimensional polyhedra, n >> 4 298 10.3 The (n - 1)-connected (n + 3)-dimensional polyedra with cyclic homology groups, n >- 4 314 10.4The decomposition problem for stable types 316 10.5 The (n - 1)-connected (n + 2)-types with cyclic homotopy groups, n >- 4 320 10.6Example: the truncated real projective spaces RPi+4/RP 327 10.7The stable equivalence classes of 4-dimensional polyhedra and simply connected 5-dimensional polyhedra 330

Chapter 11Homotopy groups in dimension 4 333 11.1 On 7r4M(A,2) 333 11.2 On 7r3(A, M(B, 2)) 344 11.3 On r4 X and f3(B, X) 347 11.3A Appendix: nilization of r4 X 354 11.4On H3(B, K(A,2)) and difference homomorphisms 356 11.5 Elementary homotopy groups in dimension 4 361 11.6 The suspension of elementary homotopy groups in dimension 4 382

Chapter 12 On the homotopy classification of simply connected 5-dimensional polyhedra 385 12.1 The groups G(q, A) 386 12.2 Homotopy groups with cyclic coefficients 392 12.2A Appendix: theories of cogroups and generalized homotopy groups 397 12.3The functor F4 402 xii CONTENTS

12.4The bifunctor r3 406 12.5 Algebraic models of 1-connected 5-dimensional homotopy types 412 12.6The case 7r3 X = 0 418 12.7The case H2X uniquely 2-divisible 418 12.8The case H2X free abelian 423

Appendix A Primary homotopy operations and homotopy groups of mapping cones 425 A.1Whitehead products 426 A.2The James-Hopf invariants 431 A.3The fibre of the retraction A V B - B and the Hilton-Milnor theorem 434 A.4The loop space of a mapping cone 441 A.5The fibre of a principal cofibration 444 A.6EHP sequences 450 A.7The operator P. 456 A.8The difference map V 463 A.9The left distributivity law 471 A.10Distributivity laws of order 3 476 Bibliography 479 Notation for categories 485 Index 487 INTRODUCTION

For each number n = 0, 1, 2,... one has the simplex A" which is the convex hull of the unit vectors a°, e1, ... e" in the Euclidean (n + 1)-space I"+' Hence i ° is a point, Ot an interval, O2 a triangle, 03 a tetrahedron, and so on: 'n, e' A2 A3

The dimension of 0" is n. The name simplex describes an object which is supposed to be very simple; indeed, natural numbers and simplexes both have the same kind of innocence. Yet once the simplex was created, algebraic topology had to emerge. For each subset a C {0, 1, ... , n} with a = (a° <

Finite polyhedra are topological spaces X homeomorphic to such substruc- tures S of simplexes 0", n >t 0. A homeomorphism S = X is called a triangulation of X. Hence a polyhedron X is just a topological space in which we do not see any simplexes. We can introduce simplexes via a triangulation, but this must be seen as an artifact similar to the choice of coordinates in a vector space or manifold.' Finite polyhedra form a large universe of objects. One is not interested in a particular individual object of the universe but in the classification of species. A system of such species and subspecies is obtained by the equivalence classes homotopy types and homeomorphism types. Recall that two spaces X, Y are homeomorphic, X = Y, if there are continuous maps f : X -> Y and g: Y --> X such that the composites fg = 1 y and t Compare H. Weyl, Philosopy of Mathematics and Natural Science, 1949: `The introduction of numbers as coordinates... is an act of violence...' 2 INTRODUCTION gf = lx are the identity maps. A class of homeomorphic spaces is called a homeomorphism type. The initial problem of algebraic topology-Seifert and Threlfall called it the main problem-was the classification of homeomorphism types of finite polyhedra. Up to now such a classification was possible only in a very small number of special cases. One might compare this problem with the problem of classifying all knots and links. Indeed the initial datum for a finite polyhedron is just a set {a1,...,ad of subsets a; c [n] as above, and the initial datum to describe a link, namely a finite sequence of neighbouring pairs (i, i + 1) or (i + 1, i) in [n], (specifying the crossings of n + 1 strands) is of similar or even higher complexity. But we must emphasize that such a description of an object like a polyhedron or a link cannot be identified with the object itself: there are in general many different ways to describe the same object, and we care only about the equivalence classes of objects, not about the choice of description. Homotopy types are equivalence classes of spaces which are considerably larger than homeomorphism types. To this end we use the notion of deformation or homotopy. The principal idea is to consider `nearby' objects (that is, objects, which are `deformed' or `perturbed' continuously a little bit) as being similar. This idea of perturbation is a common one in mathematics and science; properties which remain valid under small perturbations are considered to be the stable and essential features of an object. The equivalence relation generated by `slight continuous perturbations' has its precise definition by the notion of homotopy equivalence: two spaces X and Y are homotopy equivalent, X = Y, if there are continuous maps f: X - Y and g: Y -> X such that the composites fg and gf are homotopic to the identity maps, fg = ly and gf = 1x. (Two maps f, g: X -> Y are homotopic, f =g, if there is a family of maps f,: X -> Y, 0 < t < 1, with fo = f, fl =g such that the map (x, t) - fr(x) is continuous as a function of two variables.) A class of homotopy equivalent spaces is called a homotopy type. Using a category C in the sense of S. Eilenberg and Saunders Mac Lane [GT] one has the general notion of isomorphism type. Two objects X, Y in C are called equivalent or isomorphic is there are morphisms f : X -+ Y, g: Y - X in C such that fg = ly and gf = lx. An isomorphism typeis a class of isomorphic objects in C. We may consider isomorphism types as being special entities: for example, the isomorphism types in the category of finite sets are the numbers. A homeomorphism type is then an isomorphism type in the category Top of topological spaces and continuous maps, whereas a homotopy type is an isomorphism type in the homotopy category Top/= in which the objects are topological spaces and the morphisms are not individual maps but homotopy classes of ordinary continuous maps. The Euclidean spaces l8" and the simplexes ", n >_ 1,all represent different homeomorphism types but they are contractible, i.e. homotopy equivalent to a point. As a further example, the homeomorphism types of connected 1-dimensional polyhedra are the graphs which form a world of their own, but the homotopy types of such polyhedra correspond only to INTRODUCTION 3 numbers since each graph is homotopy equivalent to the one-point union of a certain number of circles S'. Homotopy types of polyhedra are archetypes underlying most geometric structures. This is demonstrated by the following diagram which describes a hierarchy of structures based on homotopy types of polyhedra. The arrows indicate the forgetful functors.

real algebraic sets Kaehler manifolds

I I semi-analytic sets complex manifolds Riemannian manifolds (analytic isomorphism) (complex isomorphism) (isometry)

differentiable manifolds (diffeomorphism)

polyhedra topological manifolds (homeomorphism) (homeomorphism)

I locally finite polyhedra (proper homotopy equivalence)

polyhedra (homotopy equivalence)

This hierarchy can be extended in many ways by further structures. Each kind of object in the diagram has its own notion of isomorphism; again as in the case of polyhedra not the individual object but its isomorphism type is of main interest. Now one might argue that the set given by diffeomorphism types of closed differentiable manifolds is more suitable and restricted than the vast variety of homotopy types of finite polyhedra. This, however, turned out not to be true. Surgery theory showed that homotopy types of arbitrary simply connected finite polyhedra play an essential role for the understanding of differentiable manifolds. In particular, one has the following embedding of a set of homotopy types into the set of diffeomorphism types. Let X be a finite simply connected n-dimensional polyhedron, n > 2. Embed X into a Euclidean space R"', k >_ 2n, and let N(X) be the boundary of a regular 4 INTRODUCTION neighbourhood of X c Rk+ IThis construction yields a well-defined function {X} ti {MX)} which carries homotopy types of simply connected n- dimensional finite polyhedra to diffeomorphism types of k-dimensional manifolds. Moreover for k = 2n + 1 this function is injective. Hence the set of simply connected diffeomorphism types is at least as complicated as the set of homotopy types of simply connected finite polyhedra. In dimension >- 5 the classification of simply connected diffeomorphism types (up to connected sum with homotopy spheres) is reduced via surgery to problems in homotopy theory which form the unsolved hard core of the question. This kind of reduction of geometric questions to problems in homotopy theory is an old and standard operating procedure. Further examples are the classification of fibre bundles and the determination of the ring of cobordism classes of manifolds. All this underlines the fundamental importance of homotopy types of polyhedra. There is no good intuition of what they actually are, but they appear to be entities as genuine and basic as numbers or knots. In my book [AH] I suggested an axiomatic background for the theory of homotopy types; A. Grothendieck [PS] commented:

'Such suggestion was of course quite interesting for my present reflections, as I do have the hope indeed that there exists a 'universe' of schematic homotopy types...'.

Moreover J.H.C. Whitehead [AH] in his talk at the International Congress of Mathematicians, 1950, in Harvard said with respect to homotopy types and the homotopy category of polyhedra:

'The ultimate object of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory in the same sort of way that 'analytic' is equivalent to 'pure' projective geometry'.

Polyhedra are of combinatorial nature, but they often can only be described by an enormous number of simplexes even in the case of simple spaces like products of spheres. J.H.C. Whitehead observed that for many purposes only the 'cell structure' of spaces is needed. In some sense `cells' play a role in topology which is similar to the role of `generators' in algbera. Let D'{xEW,Ilxlls1} D"=ix el",Ilxll<1}, dD" = D" _15n = S" - I be the closed and open n-dimensional disk and the (n - 1)-dimensional sphere. An (open) n-cell, n >- 1, in a space X is a homeomorphic image of the INTRODUCTION 5

0 open disk D" in X; a 0-cell is a point in X. As a set a CW-complex is the disjoint union of such cells. A CW-complex is not just a combinatorial affair since the `attaching maps' in general may have very complicated topological descriptions. Definition A CW-complex X with skeleta X ° c X' C X'- c c X is a topological space constructed inductively as follows: (a) X° is a discrete space whose elements are the 0-cells of X. (b) X' is obtained by attaching to X"-' a disjoint union of n-disks D" via continuous functionscc;: d(D;')-->X"-I, i.e.take the disjoint union X"-' - UD" and pass to the quotient space given by the identifications x - p1(x), x E dD". Each D" then projects homeomorphically to an n-cell e" of X. The map cpi is called the attaching map of a". (c) X has the weak topology with respect to the filtration of skeleta.

A CW-space is a space homotopy equivalent to a CW-complex. Homotopy types of polyhedra are the same as homotopy types of CW-spaces. The main numerical invariants of a homotopy type are `dimension' and `degree of connectedness'.

DefinitionThe dimension Dim(X) < m of a CW-complex is defined by Dim(X) X where D" is the n-dimensional disk. The 1-connected spaces are also called simply connected.

The dimension is related to homology since all homology groups above the dimension are trivial, whereas the degree of connectedness is related to homotopy since below this degree all homotopy groups vanish. It took a long time in the development of algebraic topology to establish homology and homotopy groups as the main invariants of a homotopy type. The crucial importance of homotopy groups and homology groups relies on the following results due to J.H.C. Whitehead. Theorem (A) A connected CW-space X is contractible if and only if all homotopy groups 7r"(X ), n >_ 1, are trivial. 6 INTRODUCTION

(B) A simply connected CW-space X is contractible if and only if all homology groups H (X ), n >_ 2, are trivial.

The theorem shows that homotopy groups, and in the simply connected case also homology groups, are able to detect the trivial homotopy type. In fact, homotopy groups and homology groups are able to decide whether two spaces have the same homotopy type:

Whitehead theoremLet X and Y be simply connected CW-complexes and let f: X -* Y be a map. Then f is a homotopy equivalence if and only if condition (A) or equivalently (B) holds:

(A) the map f induces an isomorphism of homotopy groups, f,,: ir,,X = for n > 2;

(B) the map f induces an isomorphism of homology groups, f* :H X = H,, Y for n>_2.

Hence both homology groups and homotopy groups constitute systems of algebraic invariants which, in a certain sense, are sufficiently powerful to characterize simply connected homotopy types. This does not mean that there is a homotopy equivalence, X = Y, between simply connected CW-spaces just because there exist isomorphisms of abelian groups H,, X = (or ir X = ?r Y) for all n. The crux of the matter is not merely that H,, X =_ H" Y, but that a certain family of isomorphisms, 4,,: H,, X = has a geometrical realization f : X -p Y. That is to say, the latter map f induces all isomorphisms 4R via the functor H,,, namely 0 = for all n. Therefore the emphasis is shifted to the following question (pointed out by Whitehead [AH] at the International Congress in Harvard (1950)).

Realization problem of J. H. C. Whitehead (A) Find necessary and sufficient conditions in order that a given set of isomorphisms or, more generally, homomorphisims ¢,,: have a geometrical realization X - Y. (B) Find necessary and sufficient conditions in order that a given set of homomorphisms, 4),,: H X -* H,,Y, have a geometrical realization X - Y.

This realization problem is far from being solved. Only for simply connected rational CW-complexes does there exist satisfying solutions by the minimal models of Quillen and Sullivan; compare Baues and Lemaire [MM]. In this book we describe new solutions for special classes of CW-complexes. As a fundamental tool we obtain the classification theorem in Chapter 3 which INTRODUCTION 7 shows that a homotopy type of a simply connected (n + 1)-dimensional CW-space X is determined in two different ways, either by the invariants

P11-1X1 HnX, Hn+1X, bn+1X, knX (*) or by the invariants

P.-1X, HnX, Hn+1X, bn+1X, $,,X. (**)

Here Pn_ 1X is the (n - 1)-type of X and bn+ 1X is the secondary boundary operator in the `certain exact sequence' of J.H.C. Whitehead [CE]. Moreover k, ,X is the Postnikov invariant of X, while 6,X is a new invariant which we call the boundary invariant of X. The classification theorem also describes all invariants (*), resp. (* *), which are realizable by spaces. All morphisms between such invariants which are realizable by maps are specified. The classification theorem yields insight into how homology groups and homotopy groups depend on each other. In fact, we classify all possible abstract homomorphisms between abelian groups,

hn' 7Tn n> which can be realized as the Hurewicz homomorphism of a space with a given (n - 1)-type; compare Theorem 3.4.7. We also show that the Hurewicz homomorphism hn can be deduced from either the Postnikov invariant knX or the boundary invariant 6,,X (see Sections 2.5 and 2.6). The following result makes clear that the Hurewicz homomorphism h,,X:vrnX->HnX has indeed a strong impact on homotopy types; compare Propositions 2.5.20 and 2.6.15.

PropositionLet X be a simply connected CW-space. Then (A) and (B) hold:

(A) the Hurewicz homomorphism hnX is split injective for all n if and only if X has the homotopy type of a product of Eilenberg-Mac Lane spaces;

(B) moreover hnX is split surfective for all n if and only if X has the homotopy type of a one point union of Moore spaces.

This result and the Whitehead theorem show that homotopy groups and homology groups are indeed the basic invariants of a homotopy type. More- over a classifying invariant of a simply connected homotopy type should determine the Hurewicz homomorphism 7Tn - Hn for each non-trivial homology group H. 1 LINEAR EXTENSIONS AND MOORE SPACES

In this chapter we describe some basic concepts used in this book. We first introduce purely categorical notions like detecting functor, linear extension of categories, and the cohomology of categories. Then we describe the properties of Whitehead's I'-functor which we shall need for the description of homotopy classes of maps between Moore spaces in degree 2. In fact, the homotopy category M2 of such Moore spaces is canonically embedded in a linear extension of categories Ext(-,f)»M2-.Ab which represents a non-trivial cohomology class in H2 (Ab, Hom( -, F)). We use Moore spaces for the definition of homotopy groups with coefficients.

1.1 Detecting functors, linear extensions, and the cohomology of categories

A letter like C denotes a category, Ob(C) and Mor(C) are the classes of objects and of maps (morphisms) respectively. The identity of an object A is 14 = 1 = id and C(A, B) is the set of morphisms A - B. The group of automorphisms of A is Autc(A) = Aut(A). An isomorphism in C is written f: A = B. An isomorphism is also called an equivalence. A natural equivalence relation - on the category C is given by an equivalence relation - on each morphism set C(A, B) such that for f, g c- C(A, B) and a, b E C(B, C) the relations f - g, a - b imply the relation of - bg. In this case we obtain the quotient category C/'- which has the same objects as C and for which a set of morphisms is the set C(A, B)/- of equivalence classes. Hence a morphism (f}: A -> B in C/- is the equivalence class of a morphism f: A - B in C. Now let = be a natural equivalence relation on a category A which is called homotopy. Then a homotopy equivalence f : A = B is the same as an isomorphism in the quotient category A/=-. The homotopy type {B} of B is the class of all objects A homotopy equivalent to B. Surfective maps, resp. injective maps, between sets are denoted by

(1.1.1) A - B,resp. A >-> B.

(The arrow >-> also describes a cofibration in a cofibration category, but the 1 MOORE SPACES 9 meaning of >-+ will always be clear from the context.) A functor A: A -+ B is full, resp. faithful if the induced maps A:A(X,Y) -> B(AX, AY) are surjective, resp. injective for all objects X, Y E A; we also write A: A -. B, resp. A: A' -> B in this case. An equivalence between categories is denoted by AB. For a functor A: A -> B let AA be the image category of A. Objects in AA are the same as in A and morphisms X -> Y in AA are the maps f : AX - AY in the image set AA(X, Y). Clearly A induces functors

A i A -AA>-+B where A is full and where i is faithful. We say that A is a quotient functor if i is an isomorphism of categories. The following notation was introduced by J.H.C. Whitehead, see for example §14 of Whitehead [CE]. (1.13) DefinitionLet A: A -* B be a functor. By the sufficiency and the realizability conditions, with respect to A, we mean the following: (a) Sufficiency: If A(f) is an isomorphism, so is f, where f is a morphism in A. That is, the functor A reflects isomorphisms. (b) Realizability: The functor is AA - B in (1.1.2) is an equivalence of categories. This is equivalent to the following two conditions (bi) and (b2). (bl) The functor A is representative, that is, for each object B in B there is an object A in A such that AA is isomorphic to B. In this case we say that B is A-realizable. (b2) The functor A is full, that is, for objects X, Y in A and for each morphism f: AX -> AY in B there is a morphism fo: X -> Y in A with Afo =f. In this case we also say that f is A-realizable. In this book the 'Whitehead theorem' is often used for checking that a functor satisfies the sufficiency condition. The proof of realizability conditions is then the hard part in classification problems. Since the sufficiency and realizability conditions appear frequently it is convenient to condense these conditions in the following definition.

(1.1.4) Definition We call A: A -> B a detecting functor if A satisfies both the sufficiency and the realizable conditions, or equivalently if A refects isomorphisms, is representative and full. Clearly, a faithful detecting functor is the same as an equivalence of categories. By a 1-1 correspondence we always mean a function which is injective and surjective. (1.15) Lemma A detecting functor A: A - B induces a 1-1 correspondence between isomorphism classes of objects in A and isomorphism classes of objects in B. 10 1 MOORE SPACES Next we consider pairs (A, b) where A is an object in A and where b: AA = B is an equivalence in B. We have an equivalence relation on such pairs given by (A, b) - (A', b') if and only if there is an equivalence g: A' -A in A with A(g) = b-'b'. The equivalence classes form the class of realizations of B denoted by (1.1.6) Reala(B) = {(A, b) I b: AA = B}/- Let {A, b} be the equivalence class of (A, b). We now consider functors which are embedded into linear extensions of categories. Such linear extensions arise frequently in algebraic topology and in many other fields of mathematics. In fact, once the reader has learnt about this concept he will recognize many examples himself and soon the usefulness and naturalness of the notion will become apparent. We first consider the classical notion of an extension of groups by modules. An extension of a group G by a G-module A is a short exact sequence of groups

0 -->A E P G-->0 i where iis compatible with the action of G, that is i(g a) = x(ia)x-' for x Ep-'(g). Two such extensions E and E' are equivalent if there is an isomorphism e: E - E' of groups with p'e = p and ei = P. It is well known that the equivalence classes of extensions are classified by the cohomology H2(G, A). Linear extensions of a small category C by a `natural system' D generalize such extensions of groups. We show that the equivalence classes of linear extensions are equally classified by the cohomology H2(C, D). A natural system D on a category C is the appropriate generalization of a G-module. Recall that Ab denotes the category of abelian groups.

(1.1.8) DefinitionLet C be a category. The category of factorizations in C, denoted by FC, is given as follows. Objects are morphisms f, g.... in C and morphisms f --> g are pairs (a, /3) for which A °-- A' fT 19 B .- B' commutes in C. Here a f 0 is a factorization of g. Composition is defined by (a', f3'Xa, )3) = (a'a, We clearlyhave(a, f3) = (a, 1X1, 13) _ (1, 8 X a, 1). A natural system (of abelian groups) on C is a functor D: FC - Ab. The functor D carries the object f to Df = D(f) and carries the morphism (a, 0): f - g to the induced homomorphism D(a,/3)=a* f3*:Df->Dafp=Dg. Here we set D(a,1) = a* and D(1, f3) =,6 *. 1 MOORE SPACES 11

We have a canonical forgetful functor zr: FC -> C°P X C so that each bifunctor D: C°P X C -* Ab yields a natural system D7r, also denoted by D. Such a bifunctor is also called a C-bimodule. In this case Df = D(B, A) depends only on the objects A, B for all f e C(B, A). As an example we have for functors F, G: Ab ---* Ab the Ab-bimodule

Hom(F, G): Ab°P X Ab Ab which carries (A, B) to the group of homomorphisms Hom(FA,GB). If F is the identity functor we write Hom(-, G) for Hom(F, G). For a group G and a G-module A the corresponding natural system D on the group G, considered as a category, is given by Dg =A for g E G and g * a = g a for a E A, g *a = a. If we restrict the following notion of a `linear extension' to the case C = G and D =A we obtain the notion of a group extension above.

(1.1.9) DefinitionLet D be a natural system on C. We say that

D+ NE P C is a linear extension of the category C by D if (a), (b) and (c) hold. (a) E and C have the same objects and p is a full functor which is the identity on objects. (b) For each f: A -* B in C the abelian group Df acts transitively and effectively on the subset p-'(f) of morphisms in E. We write fo + a for the action of a E Df on fo E p-'(f ). (c) The action satisfies the linear distributii ity law: (fo + a)(go + /3) =fogo +f* a+g*a.

Two linear extensions E and E' are equivalent if there is an isomorphism of categories e: E = E' with p' e =p and with e(fo + a) = e(fo) + a for fo e Mor(E), a e Dpf,. The extension E is split if there is a functor s: C -* E with ps = 1. We obtain the canonical split linear extension (d) D+ NCxD-.C as follows. Objects in C X D are the same as in C and morphisms X - Y in C X D are pairs (f, a) where f : X - Y E C and a E D(f ). The composition law is given by

(e) (f, a)(g, /3) = (fg,f* P+g*a). Clearly the projection C X D --* C carries (f, a) to f and the action D+ is 12 1 MOORE SPACES given by (f, a) + a' = (f, a + a') for a' E D(f). A splitting functor s yields the equivalence of linear extensions

(f) e:CXD - E given by e(f, a) = s(f) + a. We say that D+ H E 4 F is a weak linear extension if AE - F is an equivalence of categories and if D+ N E -> AE is a linear extension. In this case A is not the identity on objects, but it is easy to replace the objects in E by objects in F: for this we choose for each object X in F a realization M(X) in E so that the functor A, carrying M(X) to X, is considered as a functor which is the identity on objects. In a weak linear extension the functor A is always a detecting functor. We also consider the following maps between linear extensions D+ NE PO F (1.1.10) Id !eP, a D' + >-> E' F Here e and cp are functors with p'e = cpp and d: Df -> D,,f is a natural transformation compatible with the action +, that is e(fo+a)=e(fo)+d(a) for a E Df.

(1.1.11) LemmaIf cpis an equivalence of categories and if d is a natural isomorphism then also a is an equivalence of categories.

Let C be a small category and let M(C, D) be the set of equivalence classes of linear extensions of C by D. Then there is a canonical bijection

(1.1.12) +/r: M(C, D) - H2(C, D) which maps the split extension to the zero element, see IV §6 in Baues [AH]. Here H"(C, D) denotes the cohomology of C with coefficients in D which is defined by the nerve of C, see Definition 1.1.15 below. We obtain a representing cocycle 0, of the cohomology class {E} = q'(E) E H2(C, D) as follows. Let t be a `splitting' function for p which associates with each morphism f: A - B in C a morphism fo = t(f) in E with pfo = f. Then t yields a cocycle 0, by the formula (1.1.13) t(gf) = t(g)t(f) + O,(g, f ) with O,(g, f) E D(g, f ). The cohomology class (E) = {p,} is trivial if and only if E is a split extension. 1 MOORE SPACES 13

(1.1.14) Remark For a linear extension

D+ - E-.C (1) the corresponding cohomology class {E} = T(E) E H2(C, D) has the following classifying property with respect to the groups of automorphisms in E: for an object A in E the extension (1) yields the group extension

0 -,T- Autc(A) - Autc(A) --> 0 (2) byrestriction.Here a E Autc(A) acts on x E A = D(1A) by a x = (a-')*a* (x). The cohomology class corresponding to the extension (2) is given by the image of the class 'Ir{E} under the homomorphism H2(C, D) H2(Autc(A), A). Here i is the inclusion functor Autc(A) v C and t: i*D _A is the isomorphism of natural systems with t = (a-')*: D. -* D(1A) =X. Further results on linear extension of categories can be found in Baues [AH], Baues and Wirsching [CS] and Baues and Dreckmann [GL].

Next we define the cohomology of a category C with coefficients in a natural system D on C. In order to get cohomology groups which are actually sets we have to assume that C is a small category; by change of universe it is also possible to define this cohomology in case C is not small.

(1.1.15) DefinitionLet C be a small category and let N,,(C) be the set of sequences (Al, ... , An) of n composable morphisms in C (which are the n-simplices of the nerve of C). For n = 0 let NO(C) = Ob(C) be the set of objects in C. The cochain group F" = F"(C, D) is the abelian group of all functions

c:N"(C)-(UDg)=D (1) ge Mor(C) I with c(A,, ... ,An) E DA0 ... OA.-Addition in F" is given by adding pointwise in the abelian groups Dg. The coboundary S: F"-' -p F" is defined by the formula 00(A'I...I An) _ (AdIC(A2,..., An) n-1 + (-1)'c(AI....,AiAi,J,--.,A") i=l

+(-1)n(An)*c(A,,..., An-1). (2) For n = 1 we have (Sc)(A)= A c(A) - A*c(B) for A: A - B E N,(C). One 14 1 MOORE SPACES can check that Sc E F" for c E Frt-1 and that SS = 0. Hence the cohomology groups H"(C, D) = H"(F*(C, D), S) (3) are defined, n >_ 0. These groups are discussed in Baues [AH]. A functor 0: C' -> C induces the homomorphism

(1.1.16) ¢*: H"(C, D) --> H"(C', 4*D) where O*D is the natural system given by (4*D)f=DD(f). On cochains the map 4* is given by the formula

where (A', ... , A;,) E N"(C'). In (IV.5.8) of Baues [AH] we show:

(1.1.17) PropositionLet 0: C - C' be the equivalence of categories. Then (A* is an isomorphism of groups.

A natural transformation T: D -p D' between natural systems induces a homomorphism

(1.1.18) T H"(C, D) -> H"(C, D') by (T*fXA1...... ")=TAf(A1,...,A")where rA:D,-->DD with A=A1o oA,, is given by the transformation T. Now let

r D" be a short extract sequence of natural systems on C. Then we obtain as usual the natural long exact sequence

(1.1.19) --> H"(C, D') - H"(C, D) - H" (C,D") _ isH" + 1(C, D') where /3is the Bockstein homomorphism. For a cochain c" representing a class (c") in H"(C, D") we obtain /3(c") by choosing a cochain c as in (1) of Definition 1.1.15 with rc = c". This is possible since T is surjective. Then i-1Sc is a cocycle which represents /3{c" }.

(1.1.20) Remark The cohomology of Definition 1.1.15 generalizes the cohomology of a group. In fact, let G be a group and let G be the corresponding category with a single object and with morphisms given by the elements in G. A G-module D yields a natural system b: FG --> Ab by Dg = D for g e G. 1 MOORE SPACES 15

The induced maps are given by f *(x) =x and h*(y) = It y, f, It E G. Then the classical definition of the cohomology H"(G, D) coincides with the definition of H"(G, D) = H"(G, D) given by Definition 1.1.15.

1.2 Whitehead's quadratic functor IF

We describe the universal quadratic functor F: Ab --> Ab which was introduced by J.H.C. Whitehead [CE] and which also was considered by Eilenberg and Mac Lane [II]. The functor F is characterized by the following property: a function 17: A -, B between abelian groups is called a quadratic map if 77( - a) _ q(a) and if the function A X A --+ B with (a, b) H q(a + b) - -q(a) - q(b) is bilinear. For each abelian group A there is a universal quadratic map (1.2.1) y:A-'F(A) with the property that for all B and all quadratic maps 71: A -p B there is a unique homomorphism 170 : I'(A) -* B with 71'y=f. Now r is a functor since a homomorphism cp: A -- B yields the quadratic map yqp which induces a unique homomorphism F(op) = (yc)° such that the diagram

-. F(A) - - T(B)

1 1 A B w commutes. We have the following examples of quadratic maps. Let a,: A -A (9 Z/2 be given by o,(a) = a ® 1. Then it is easily checked that ais quadratic. Therefore we obtain the canonical surjective homomorphism

(1.2.2) withQy=a .

We consider the function Ho: A -+A ®A with H0(a) = a ® a. Clearly, H. is quadratic and yields the canonical homomorphism (1.2.3) H:I'(A)-*A®A with Hy=Ho.

The cokernel of H is the exterior square A A A =A ®A/(a ®a - 0). Next we obtain by the quadratic map y the bilinear pairing

(1.2.4)

[,] _ [1,1]: A ®A -' I (A) with[a, b] = y(a + b) - y(a) - y(b). 16 1 MOORE SPACES

We write[f, g] = [1,1X f ®g): X ®Y -*A ®A --* F(A) wheref: X -+A, g: Y->A are homomorphisms. Clearly, we have [a, b]=[b, a] and

(1.2.5) r[a,b]=0 and H[a,b]=a®b+b®a. Moreover, the sequence

(1.2.6) A ®A [I''1 r(A) A ®7L/2 ---> 0 is exact and natural in A. By considering the equation [a + b, c] _ [a, c] + [b, c] we see that the following relations are satisfied in F(A):

J y(-a) = y(a) y(a+b+c)-y(a+b)-y(b+c)-y(a+c)+y(a)+y(b)+y(c)=0)

(*)

We can construct the group I'(A) as follows. Consider the map y: A -* A where A is the free abelian group generated by the underlying set of A. The map y is the inclusion of generators. We set r(A) =A/R where R denotes the relations (*) with y replaced by y. Now y is the composite A -> A --' A/R of y and of the quotient map. One easily checks that this composition has the universal property in (1.2.1). For a direct sum A ® A' we have the isomorphism

(1.2.7) I'(A (&A') = r(A) ®A (&A' e I'(A') which is given by F(i), r(i'), and [i, i'], where i, i' are the inclusions of A and A' into A ® A' respectively . A similar result is true for an arbitrary direct sum where I is an ordered set:

1'( A)= ® F(A;) ®® A. ®AJ. i

Moreover, I' commutes with direct limits of abelian groups. If A = 71 then F(A) = 71 is generated by y1. This shows that for a free abelian group A, also F(A) is free abelian. If B is an ordered basis of A then (y(b) I b E B) U ([b, b'] I b < b'; b, b' E B) is a basis of I'(A). For an arbitrary abelian group we obtain a presentation of r(A) by the following crucial result: d-' (1.2.8) LemmaLet C D -' A - 0 be an exact sequence of abelian groups. Then the sequence r(C) ®C ®D -L F(D) -3 r(A) -> 0 is exact. Here d is defined by I= (F(d),[d,1]) where 1 is the identity of D. 1 MOORE SPACES 17

If we set a = b = -c in (*) above we get y(a) - y(2a) + 3y(a) = 0 and therefore y(2a) = 4y(a). More generally we get y(na) = n2y(a). By definition we see 2y(a) = [a, a]. Using these equations we derive from the lemma:

(1.2.9) CorollaryFor a cyclic group 7L/n = Z/nZ we have r(7L/n) =7L/(n22,2n) where (n2,2n) is the greatest common divisor. The group is generated by y(l) where 1 is a generator of 7L/n, 1 = 1 + n7L.

Hence the surjective homomorphism (1.2.10) H: I'UL/n) - 7L/n ® 7L/n = 77/n has kernel Z/2 if n is even and is an isomorphism if n is odd. Clearly, H is surjective since Hy(1) = 10 1 is a generator. We derive from the universal property of y that the following diagram commutes since [1,1]Hy(a) = [a, a] = 2y(a).

H A 0A [I I'A I'A 2 Here 2 is the multiplication by 2. This shows:

(1.2.11) PropositionLet A be an abelian group such that multiplication by 2 is an isomorphism on A. Then multiplication by 2 on FA is an isomorphism and H: I'A -A ®A is injective and admits a natural retraction, namely (1/2) [1, 1].

Proof Since 2 is an isomorphism, I'A -+ I'A is also an isomorphism. Since 4 = is also an isomorphism on I'A.

We finally describe an important example in topology. It is a classical result of J.H.C Whitehead [CES] that the Hopf map 712: S3 -> S2 induces a quadratic function

(1.2.12) 772:1r2X--+7r3X, 11 (a)=ail, between homotopy groups of a space X. This function satisfies the left distributivity law

772 (a+ J3) - 712* (a) - T12(13) = [a, J3] where [a, (3 ] E'T3 X is the Whitehead product of a, )SE 7r2X. Clearly the induced homomorphism 17 = (71z) °, (2.1.13) 77:F7r2X-*7r3X with7r(ya)=rr2(a), carries the bracket [a, 13 ] E r7r2 X to the Whitehead product [a, (3 ] E 7r3 X. 18 1 MOORE SPACES

1.3 Moore spaces and homotopy groups with coefficients

We start with the definition of a Moore space. Let n > 2. A Moore space of degree n is a simply connected CW-space X with a single non-vanishing homology group of degree n, thatisH;(X,7L) = 0 fori # n. We write X = M(A, n) if an isomorphism A = H"(X,7L) is fixed. Let Mn be the full homotopy category of all Moore spaces of degree n. We have the homology functor H": M" -* Ab which carries M(A, n) to the abelian group A. The (n - 1)-fold suspension of a pseudo projective plane Pq = S' Uq e2 is a Moore space of the cyclic group 7L/q = 7L/q7L, that is Y."-'Pq = M(7L/q, n). Clearly a sphere S" is a Moore space S" = M(7L, n).

(13.1) Lemma The functor M" - Ab is a detecting functor, that is, for each abelian group A there is a Moore space M(A, n), n >- 2, the homotopy type of which is well defined by (A, n). Moreover, for each homomorphism cp: A - B there is a map gyp: M(A, n) -> M(B, n) with H";P = cp.

One has to be careful since for two Moore spaces X, Y of type M(A, n) there is no `canonical' choice for the homotopy equivalence X = Y, since the homotopy class ;P is not uniquely determined by gyp. We can easily construct d M(A, n) as follows. Choose for A a short exact sequence C N D --.A where C and D are free abelian. Then d yields, up to homotopy, a unique map d: M(C, n) --> M(D, n) the mapping cone of which is M(A, n). For M(C, n) and M(D, n) we can take one-point unions of n-spheres. This shows that M(A, n) can be represented by a CW-complex with cells only in dimension n and n + 1. The suspension of a Moore space of degree n is a Moore space of degree (n + 1), that is Y.M(A, n) = M(A, n + 1). Also a Moore space of degree 2 has the homotopy type of suspension M(A, 2) = Y. MA where MA for example can be chosen to be the mapping cone of a map d': Mc --> MD where Mc and MD are one-point unions of 1-spheres and Id' = d. The homotopy type of MA is not determined by A.

(13.2) DefinitionLet U be a space with base point *. The homotopy set of base-point preserving maps (1.3.3) it"(A; U) = [M(A,n),UJ (n>:2) is called a homotopy group with coefficients in A. For n > 3 this is an abelian group. Also ir2(A; U) has a group structure which, however, depends on the choice of MA. These homotopy groups are covariant functors in U. They are not contravariant functors in A; but they are contravariant functors on the homotopy category M' of Moore spaces of degree n. The following proposition is the `universal coefficient theorem' for homotopy groups. 1 MOORE SPACES 19

(13.4) PropositionFor n > 2 there is the central extension of groups

Ext(A, 7r,1U) Nit"(A; U) Hom(A, it"U) which is natural in U and which is natural in A in the following sense. Let ;p: M(A, n) -' M(B, n) be a map with H" 4P = cp: A -> B. Then we have 0 cp* _ ;P*0 and A;p* = 9*µ.

Proof For the mapping cone M(A, n) of the map d above we have the exact cofibre sequence of groups Hom(D,7r" ,) -Hom(C,7r"+i) -> [M(A,n),U] Hom(D, 70 ---> Hom(C, 7r"), where 7r" = 7r"(U). For n = 2 the extension is central by (11.8.26) Baues [AH]. 11

The universal coefficient sequence is compatible with the suspension functor 1. In fact we have the following commutative diagram of homomorphisms between groups, n > 2. Ext(A,7r"+,U) H 7r"(A; U) - Hom(A, 7r"U)

(1.3.5) Ext(A,ir"+z ,U). This follows easily by the naturality in U if we consider U-> J1 XU where (1IU is the loop space of 1U. We now consider the categories M' of Moore spaces. The suspension functor I yields the sequence of functors

E (1.3.6) M'-.M3-M4--. which commute with the homology functor, that is H"+,Y.= H": M" - Ab. The Freudenthal suspension theorem shows that I is full on MZ and that the tlinctor 1: M' -> M"+ 1is an equivalence of categories for n > 3. Therefore it ik enough to compute M2 and M3. Let F be the quadratic functor of J.H.C. Whitehead and let y: A -+ I'(A) be the universal quadratic function. Then we have the suspension map o : F(A) -*A ®1/2 witho (ya) = a ® 1. We define f Dr n >_ 2 the functor F,,': Ab - Ab by

(1.3.7) ri(A) =J r(A) for n = 2 A®71/2 forn3. /e have a natural isomorphism

17: 7r"+,M(A,n) 20 1 MOORE SPACES which, for a E A = ir" M(A, n), is defined as follows. For n = 2 the isomorphism y(a) to riz (a) where 'rig: S3 - S2 is the Hopf map; in this way we identify riz with the universal quadratic map '2 (1.3.8) y: A = 7r2M(a,2) 7r3M(A,2) =1'(A). Moreover, for n >_ 3 the isomorphism carries a ® 1 to ri,*a where ri" = I'- 2,n2: S"+ ` -> S"is the suspended Hopf map. The suspension I on 7r3M(A,2) is now identified with (1.3.9) a-: F(A) = Tr3M(A,2)-L a4M(A,3) =A ® Z/2. As a corollary of the universal coefficient theorem we get

(13.10) CorollaryFor n >_ 2 one has the binatural central extension of groups

N [M(A,n),M(B,n)] -µ.Hom(A,B) where H" is given by the homology functor. We set P + a = + 0(a) for (p:A -> B. Then the `linear distributwity law'

(41 is satisfied with 0: B -> C E Ab and a E Ext(A, r'B), 0 E Ext(B, r,,' 0.

The corollary shows that we have a linear extension of categories (n >_ 2) (1.3.11) M"-Ab; compare also (V.3a) in Baues [AH]. This implies that the group of homotopy equivalences ( (X) for X=M(A,n) is embedded in the extension of groups

(1.3.12) Ext(A, r'A)>'*) (C(X) -. Aut(A). Here Ext(A, r A) is an Aut(A )-module by cp a = p * (gyp-' )* (a ). The inclusion homomorphism 1 + is defined by 1 '(a) = 1 + a where 1 is the identity of M(A, n). The linear distributivity law shows that 1+ is actually a homomorphism. The extensions (1.3.11) and (1.3.12) in general are not split, see Baues [AH]. Next we obtain, forgyp: A -b B and a E [M(A, n), M(B, n)],a e Ext(A, F, B), the induced function

(1.3.13) (gyp + a)*: 77r"(B, U) - 17"(A, U) which satisfies the formula, x E ir"(B, U), (ijp + a)* (x) = ;p(x) + Aa * µ(x). 1 MOORE SPACES 21

Here defined by the commutative diagram B b ''

b) ° %'B where y is the universal quadratic function for n = 2 and is o-y for n > 3.

(13.14) DefinitionFor n >_ 3 let ir,(A, X) be the kernel of the homomorphism A Hom(A,?rX) t1 - Hom(A,Trn+1X).

Then (1.3.13) shows that rr is actually a well-defined bifunctor

ir,,': Ab°P X Top/= --> Ab together with a natural short exact sequence

Ext(A, rri+1X) X) -Hom(A, ir,,X).

Here denotes the kernel of 71,*: ir (X) 1T,, I(X).

1.4 Suspended pseudo-projective planes

Pseudo projective planes, P11 are the most elementary 2-dimensional CW- complexes. They are obtained by attaching a 2-cell e2 to a 1-sphere S' by an attaching map f : S' -> S' of degree f >_ 1, that is

(1.4.1) Pf=S'Ufe2=D/-.

Here D is the unit disk of complex numbers with boundary S' = dD and with base point * = 1. The equivalence relation - on D is generated by the relations x - y p xf = yf with x, y E S1. Clearly P2 = 118 P2 is the real projective plane. We obtain, for each pair 17) of natural numbers with g6 = rlf, a map

(1.4.2) rt:Pf-->Pgby for xED.

The induced map -rr,T : 7L/f = rr Pf -+ Tr1 Pg = 7L/g on fundamental groupsis given by the numberq = ge/f which carries the generator 1 E 7L/f with 1 = 1 + f7L to r) 1 E 7Lg. We call the homotopy class ofT£ in Top* a principal 22 1 MOORE SPACES map between pseudo-projective planes. We point out that the homotopy class of Tr is not determined by ?r1('rt ); see III Appendix B in Baues [CH] where we compute the set [Pf, Pg]. We consider the suspensions (1.4.3) I"-'Pf=S" Uf e"+' =M(7L/f,n) of pseudo-projective planes, n >_ 2, which are Moore spaces of cyclic groups.

(1.4.4) TheoremFor (p E Hom(7L/f,71/g) there is a unique element 45 = B2(0 E [P1, ' Pg ] which induces cp = H2and which is the suspension of a principal map Pf -> Pg. Moreover for n > 3 there is a unique element ip = Bn(cp) E [y"'P1, 'Pg] which induces tp = H"and which is the (n - 1)-fold suspension of a map Pf -> Pg.

This crucial fact is proved in III Appendix D of Baues [CHI. Let P" be the full subcategory of Top*/= consisting of the sphere S" and the spaces In-1pf, f > 1. Let (1.4.5) H":P"_Cyc be the homology functor where Cyc is the full subcategory of cyclic groups i/n, n > 0, in Ab with 71/0 = Z.

(1.4.6) CorollaryFor n >_ 2 the homology functor in (1.4.5) admits a splitting functor B":Cyc--+ P" with H" B" = 1.

For the proof of the corollary we only observe that the composition of principal maps between pseudo-projective planes is principal. Hence the corollary is an immediate consequence of the theorem. For 4n-1Pf= S" Uf en+'we have the inclusion of the bottom sphere i and the pinch map q such that (1.4.7) S" ` " 'Pf 9 Sn+1 is a cofibre sequence. The function Bn: Hom(7L/f,Z/g) --* [In-1Pf, In-'P5] in Theorem 1.4.4 is not additive. But we have the following rule:

(1.4.8) TheoremFor cp, gyp' E Hom(7L/f,7L/g) we have (p') =Bn(cc) +B,,(cp') +o(w, v') L1(V' V') _ (f(f - H e r e gyp,, Vi a r e numbers w i t h o(l) _ (pt1 and qp'(l) = (p'1 and 71n: Sn+' -Sn is the Hopf map. In particular we get Bn(rcp) = Zr(r - cp). 1 MOORE SPACES 23

This result, together with the central extension of groups (see Proposition 1.3.4 and 1.3.7)

(1.4.9)Ext(7L/f,I','Z/g) H [Y."-'Ps,Y."-'PgJ y Hom(7L/f,71/g) determines completely the group structure of [' -'Pf, I" -'Pg ]Here the kernel of H" is a cyclic group generated by the element iq"q; recall that forthecyclicgroups71/gwe have1'(7L/g) = Z/(g2,2g) sothat Ext(71/f, r(71/g)) = 71/(f, g2, 2g). Here (g2, 2g) and (f, g2, 2g) denote the greatest common divisors.

(1.4.10) CorollaryFor all f, g the group [1Pf, 5,Pg] is abelian. Let a, b be defined by f = 2°fo and g = 2bgo where fo and go are odd. Then the homomorphism (n > 2) H": [I" 'Pf," 'Pg] -" Hom(7L/f,7L/g) = 7L/d has an additive splitting of abelian groups if and only if (a, b) # (1, 1). Moreover ford=(f,g), c=(f,g2,2g), ande=(f,g,2) we have

7L/d ®7L/c for (a, b) 0 (1, 1) [IPf,IPgI 7L/2d ®71/(c/2)for (a, b) _ (1,1)

(7L/d ®7L/e [12Pf'1'2p ]- for (a, b)(1,1) 7L/2d for (a, b) = (l, 1).

Compare also III Appendix D in Baues [CH] and Barratt [TG]. A restriction of the linear extension (1.3.11) for M" yields the linear extension NP"_Cyc which is split by B" for n > 2. Hence Definition 1.1.9(f) yields the equivalence

(1.4.11)

Moreover for n > 3 Theorem 1.4.8 determines P" as a pre-additive category. We use this fact in the following section.

1.5 The homotopy category of Moore spaces M', n >_ 3

In Section 1.4 we computed completely the homotopy category P" ofsus- pended pseudo-projective planes. Here we use this result for the computation of the homotopy category M" of Moore spaces. We recall the following notation: 24 1 MOORE SPACES

(1.5.1) Definition A category P is pre-additive or a ringoid if all morphism sets P(A, B) are abelian groups such that the composition is bilinear. The category of matrices over P is given as follows. Objects are tuples (A1,. .. , An) of objects in P and morphisms from (A1, ... , A,,) to (B1, ...,B,n) are matrices M=(MMiEP(A;,BJ)Ii=1,...,n;j=1,...,m). Composition is defined for N: (B1..... (C...... Cs) by

(NM)ki= F,M NkJoM,EP(Ai,Ck). i=1

The category of matrices over P is also called the additive completion Add(P). A category A is called additive if A is pre-additive and if finite sums exist in A. Each additive functor F: P - A has a unique additive extension F: Add(P) --> A which carries (A 1, ... , A,,) to the sum of FA1,..., FA,, in A. Recall that a functor between ringoids is additive if it is a homomorphism on morphism sets.

We consider the full homotopy category FM' (n >_ 2) which consists of Moore spaces M(A, n) where A is a finitely generated abelian group. For each such group we have a direct sum decomposition (1.5.2) A=7L/a1 ®... ®77/ar,a;>0, of cyclic groups. Associated with this isomorphism there is a homotopy equivalence M(A, n) =Y"-1(P0 v ... V pa ) where P,, = S1 U,, e2 is a pseudo-projective plane if n > 0 and where P,, = S1 for n = 0. This leads to the following result:

(1.53) PropositionFor n >_ 3 the category FM" is equivalent to the category of matrices over the pre-additive category Pn.

Proof The inclusion P' c FM" yields the additive extension Add(P") -' FM" which is an equivalence of categories.

There are certain subcategories of Mn which have a simple algebraic characterization. We use the ring R,=Iyn-1p,,yn-1P,1,n>3. The ring structure is given by composition and addition of maps. In Section 1.4 we computed the rings R,. 1 MOORE SPACES 25

(1.5.4) CorollaryThe full subcategory in M" (n >- 3) consisting of Moore spaces M(A, n) for which A is a finitely generated free 7L/t-module is equivalent to the category of finitely generated free R,-modules.

(1.5.5) Remark Let FAb be the full subcategory of Ab consisting of finitely generated abelian groups. Then we obtain the non-split linear extension (n>-2) H FM" - FAb (1) which is the restriction of (1.3.11). Using (1.1.12) we thus have the non-trivial element

0 (FM") EHZ(FAb,Ext(-,1:"')) (2) which, however, restricted to the subcategory Cyc C FAb is trivial. On the other hand M. Hard has shown that the cohomology -7L/2 (3) is a cyclic group of order 2. Hence (FM') is the generator of the group so that the linear extension (1) is, up to equivalence, the unique extension of FAb by Ext(-, which is not split. For n >- 2 this is a kind of fancy characterization of the category FM'. Using Proposition 1.5.3 one can compute a cocycle representing the cohomology class (FM"), n >- 3; for n = 2 we shall compute such a cocycle below.

1.6 Moore spaces and the category G In this section we describe an algebraic representation of the homotopy category of Moore spaces Mn, n >- 3. Using homomorphisms between certain abelian groups G(A) we define an algebraic category G and we describe an equivalence of additive categories Mn = G for n z 3. This then leads to the computation of the homotopy group ir"(A, X), n3, with coefficients in A in terms of the operator-q" ' 1r"(X) -* 1r"+ (X) induced by the Hopf map 71". We identify Ext(7L/q, A) = A ®7L/q = A/qA Hom(7L/q, A) =A *7L/q = Ker(A -- »A). The identification is natural in A, but clearly not natural in 7L/q. Moreover we shall use the following natural transformation of functors g=g9:Ext(A,B)->Hom(A*7L/q,B®7L/q) (1.6.2) Ig(a)(x) =x*(a). 26 1 MOORE SPACES

Here an element x EA *71/q = Hom(71/q, A) induces the homomorphism x*: Ext(A, B) ---> Ext(Z/q, B) = B ® 7L/q.

(1.6.3) LemmaIf A is a free 71/q-module then gq above is an isomorphism.

Compare (IX.52.2) in Fuch [1]. We now consider, for n >_ 3, the abelian group (1.6.4) G(A) = irn(71/2, M(A, n)) which does not depend on n >_ 3 since we have the suspension isomorphism. The extension

A ® 71/2 >G(A) A * Z/2, (1) given by (1.6.1) and Corollary 1.3.10, thus yields an extension element

{G(A)} E Ext(A * Z/2, A (9 71/2) = Hom(A * l/2, A ® Z/2). (2) The corresponding homomorphism

GA =g(G(A)): A *71/2 -A ® 71/2 (3) carries x to A-'(2x) where x E G(A) is an element with Ai =x.

(1.6.5) PropositionThe homomorphism GA coincides with composition GA:A*Z/2CA -»A(9 71/2, of the inclusion and projection.

In particular we have G(A ® B) - G(A) ® G(B) and 71/4 ifq2mod4, G(l/q)- 71/2®71/2if q0mod4, 0 otherwise. This yields G(A) for each finitely generated abelian group A.

Proof of Proposition 1.6.5For z E G(A) we have 2 x = (2 id)*x where 2 id is given by the homotopy commutative diagram In- IP zid z En - 'Pz

19 I Sn+ 1 Sn. 41 1 MOORE SPACES 27

Here n, is the Hopf map. Now the definition of 0 via q and rl" in (1.3.7), and (1.6.03) yields the result.

We use the extension for the group G(A) above in the definition of the following category.

(1.6.6) Definition of the category G Objects are abelian groups. A map from A to B is a pair ((p, ip) with p E Hom(A, B) and 4 E Hom(GA,GB) such that the following diagram commutes:

A®7L/2>->G(A)-*A*71/2 1'P®I to 1'.1 B ®7L/2 >-> G(B) -.B *71/2

We call (cp, qr) a proper map G(A) -), G(B). Let G(A, B) be the set of all proper maps from G(A) to G(B). Then the naturality of the universal coefficient sequence yields a function

G: [M(A, n), M(B, N)] --* G(A, B) G(yp)= where rp = H";P and ;P* = it"(71/2, gyp). Clearly, G is a functor; in fact:

(1.6.7) TheoremThe functor G: M' -' G is an equivalence of additive categories (n >_ 3) and there is a commutative diagram of linear extensions of categories

E" N Mn HAb

-pr FN G Ab

Here g is the isomorphism

g: E"(A, B) = Ext(A, B (9 71/2) = Hom(A *71/2, B ®7L/2) = F(A, B) given byg2 above. The functor pr: G -p Ab is the projection (cp, r(r) >- p and the action F N G is given by (cp, 40 + P(gyp, 4 + AjS µ) for J3 E F(A, B).

(1.6.8) CorollaryLet n >_ 3 and let A be any abelian group. Then the group of homotopy equivalences of the Moorespace M(A, n) is the group of proper automorphisms of G(A). 28 1 MOORE SPACES

Proof of Theorem 1.6.7It is enough to show that the following diagram commutes Ext(A,B®Z/2) -g-iHom(A *Z/2,B(&71/2)

[M(A, n), M(B, n)] Hom(G(A),G(B)) where ;P+( a)= p +a is given by the universal coefficient theorem and where we set ;P * (/3) =* + A 13 µ. Now we have,by the distributivity law for Mn' the following equation, where x E [I"-'P2, M(A, n)] = G(A):

Theorem 1.6.7 yields a complete algebraic description of the additive category M', n >_ 3, as an extension of Ab. Such a simple description is unfortunately not available for the category M2 of Moore spaces of degree 2. We now consider, for n >_ 3, the functorial properties in A of the homotopy groups 7r" (A, X) = [M(A,n),XI with coefficient in an abelian group. Using the equivalence of categories G in Theorem 1.6.7 we obtain, for each pointed space X, a functor

(1.6.9) G°p = (M-)'p , Ab which carries A E G to the group 7r"(A, X). For the computation of this functor we introduce the following notation.

(1.6.10) DefinitionLet A, 7r, 7r' be abelian groups and let -0: 7r ® 71/2 --p 7r' be a homomorphism. We define the abelian group G(A,71) by the following push-out diagram in Ab in which rows are short exact.

Fi Hom(A*Z/2,7r(9 71/2) 11G(A,7r)- Hom(A,7r)

fl Ext(A, 7r ® 1/2) push

117 1 II Ext(A,7r') 0=G(A,77)-µ Hom(A,7r) The top row is given by the abelian group G(A, 7r) of morphisms A ---> 7r in the category G (see Definition 1.6.6), with A (q, ir) = (p and A(0) = (0,A)3µ) (see (1.6.01)). The isomorphism on the left-hand side is defined in (1.6.4)(2). The diagram is in the obvious way functorial in A E G and hence we obtain 1 MOORE SPACES 29 by G(A, 71) a functor G(-, -q): G°P -> Ab. This functor is used in the next result for the computation of the functor (1.6.9).

(1.6.11) TheoremLet n >- 3 and let X be a pointed space and let

rt= nn* :1rn(X) ® 71/2 -> am+,(X) be induced by the Hopf map -qn. Then one has an isomorphism of groups

irn(A,X) =-G(A,71) which is natural in A E G and for which the following diagram commutes:

Ext(A, irn+ IX) N -rrn(A, X) - Hom(A,

Ext(A, Tr,,1X) ° G(A, rl) -3 Hom(A, nr,, X)

Here the top row is the universal coefficient sequence in Proposition 1.3.4 and the bottom row is given by the diagram in Definition 1.6.10. We point out that G(A, -q) is not functorial in r) and that the isomorphism of Theorem 1.6.11 is not natural in X.

(1.6.12) CorollaryLet n >- 3 and assume that irn(X) and A are finitely generated abelian groups. Then the (0, µ)-extension for irn(A, X) is split if and only if one of the following three conditions is satisfied:

(a) A has no direct summand 1/2; (b) 7rn(X) hasno direct summand 71/2; (c) each elementa e irn(X) generating a direct summand Z/2 satisfies q(a) _ ao71n=2a' for some a'Eirn+,(X).

Hence, if (a), (b), or (c) hold, one has an isomorphism of abelian groups

Ext(A,ir,, 1X) which, however, is not natural in Aor in X.

Proof of Corollary 1.6.12If (a) or (b) hold the top row in the diagram of Theorem 1.6.11 is split;if (c) holds the bottom row in the diagram of Theorem 1.6.11 is still split. 30 1 MOORE SPACES

Proof of Theorem 1.6.11 We may assume that X is a connected CW-space. Let f : Y - X be the (n - 1)-connected cover of X; this is the fibre of the Postnikov map X --> P X, see Section 2.6. Then f induces isomorphisms ir,(f) for i > n. For Tr = ir X we can choose a map g: M(-, n) -, Y which induces the identity in homology. Such a map g exists since Y is (n - 1)-connected, n >_ 3. The map f induces an isomorphism

and g induces the commutative diagram

Ext(A, rr (9 71/2) 7T,,(A, M(7r, n)) -» Hom(A, ir) IExt(A,, ) Ext(A,7r,,,X) N -*Hom(A,Tr)

where ,1= 71,* is induced by the Hopf map. Since the rows are short exact this is a push-out diagram of abelian groups. Therefore Theorem 1.6.11 is a consequence of the isomorphism G in Theorem 1.6.7. 2 INVARIANTS OF HOMOTOPY TYPES

The classical algebraic invariants of a simply connected homotopy type {X) are the homotopy groups ir (X) and the homology groups H (X ), n >_ 2. They are connected by the Hurewicz homomorphism (1) which is embedded in Whitehead's certain exact sequence b" ,X- b,X H 1(X) I' X ->. (2)

It is well known that neither homotopy groups ar*(X) nor homology groups H* (X) suffice to determine the homotopy type of X. However a simply connected space X is contractible if and only if all homotopy groups 7T, ,X vanish, or equivalently if all homology groups H X vanish. Moreover one has the following facts which show that the Hurewicz homomorphism is indeed significant for the characterization of homotopy types.

Proposition 2.5.20 A simply connected space X is homotopy equivalent to a product of Eilenberg-Mac Lane spaces if and only if is split injective for all n.

Proposition 2.6.15 A simply connected space X is homotopy equivalent to a one point union of Moore spaces if and only if is split surfective for all n.

In addition to the Hurewicz homomorphism (1) and Whitehead's exact sequence (2) we have to study deeper invariants of a homotopy type. There are, on the one hand, Postnikov invariants or k-invariants which are related to homotopy groups; they are nowadays explained in many textbooks on homotopy theory. On the other hand, we introduce new invariants of a simply connected homotopy type which we call boundary invariants. They are related to homology groups similarly to the way Postnikov invariants are related to homotopy groups. The duality between Postnikov invariants and boundary invariants is striking. The main results in this chapter describe properties of Postnikov invariants and boundary invariants respectively. Our results on boundary invariantsare new and also some of the properties of k-invariants described in this chapterseem to be new. We use Postnikov invariants and boundary invariants for the classification of homotopy types. The fundamental classification theorem basedon these invariants is obtained in Section 3.4. We also consider unitary invariants ofa homotopy type which are introduced by G.W. Whitehead [RA]. 32 2 INVARIANTS OF HOMOTOPY TYPES 2.1 The Hurewicz homomorphism and Whitehead's certain exact sequence

We consider the Hurewicz homomorphism which is embedded in Whitehead's certain exact sequence and we define the operators in the sequence. We also fix some notation for homotopy groups, (co)homology groups, and chain maps. Let Top* be the category of topological spaces with base point * and of base-point preserving maps. A homotopy H: f = g in Top* is given by a map H: 1 * X - Y with Hi o =f, Hit = g. Here 1 * X = I X X/I X *is the reduced cylinder on X (given by the unit interval I = [0, 1]) and i,: X -+ I * X is the inclusion with i,(x) = (t, x) for t E I, x E X. Let

(2.1.1) [X, Y] = Top*(X, Y)/= be the set of homotopy classes of maps f: X - Y in Top*. This is the set of morphisms {f ): X -> Y in the quotient category Top*/=. We often denote the homotopy class {f} simply by f. We have the trivial map 0: X --> * -s Y which represents 0 E [X,Y]. The cone of X is CX = I* X/i1X and the suspension of X is IX = CX/i°X. The n-sphere S" satisfies ES" = S"+ 1 and homotopy groups are given by (n >_ 0)

zr"(X)= [S", X] (2.1.2) ir"+1(Y,X)=[(CS",S"),(Y,X)] where (Y, X) is a pair in Top*. We have the long exact sequence of homotopy groups (n >_ 0)

i (2.1.3) 7rn+1X `''r"+1Y J'7T"+1(Y,X)--0ir"X s7r"Y. Here a is the restriction and j is induced by the quotient map (CS", S") - (S"+ 1, * ). Clearly i is induced by the inclusion X c Y. The exact sequence is natural with respect to pair maps (X,Y) - (X',Y') in Top*. We are mainly interested in CW-spaces also termed spaces. These are spaces which have the homotopy type of a CW-complex in Top*/=. Let X be a CW-complex with skeleta X". A map F: X -+ Y between CW-complexes is cellular if F(X") c Y". Let CW be the following category. Objects are CW-complexes X with trivial 0-skeleton X ° = * and morphisms are cellular maps F: X --- Y. The objects of CW are also called reduced CW-complexes. The cylinder 1* X of a CW-complex X is again a CW-complex with skeleta

(2.1.4) (I*X)"=X"UI*X"-1 UX

We call a cellular map H: I* X -* Y a 1-homotopy and we write H: f =1 g. 2 INVARIANTS OF HOMOTOPY TYPES 33

Moreover we call H a 0-homotopy if Hi, is cellular for all t E I; in this case o o we write H: fg. The natural equivalence relations and 1 yield the quotient functors

o (2.1.5) CW -> CW/ -> CW/i= CW/= between the corresponding homotopy categories. The following cellular approximation theorem shows that actually CW/= = CW/ this is a full subcategory of Top*/=.

(2.1.6) Cellular approximation theoremLet X0 be a subcomplex of the CW-complex X and let f : X - Y be a map such that f I X0 is cellular. Then there exists a cellular map g: X -> Y with g I X0 = f I X0 and with f = g rel X0.

For this recall that a homotopy H: f = g is a homotopy rel X0 (resp. under X0) if f I X0 = (Hi,) I X0 for all t E I. We now associate with a CW-complex X the cellular chain complex C* X; this is the chain complex defined by the relative homology groups (2.1.7) C"X = H"(Xn, X"-') with the boundary d=dn:Hn(Xn,X'-1) a.Hn 1(Xn-1)! H" 1(Xn-I,X"-2) given by the triple (Xn,Xn-',Xn-2). Let Chains be the following category. Objects are chain complexes C = (Cn, d, n E Z) of abelian groups and morphisms F: C' --> C are chain maps. Two such chain maps are homotopic, F = G, if there exists a homomorphism a : C' --> C of degree +1 with d a + ad = -F + G. The chain map F is a homology isomorphism if F induces an isomorphism F* : H* C' a H,k C. Here the homology of C is defined by the quotient group (2.1.8) H"C = Zn/Bn where Zn = kernel{dn: Cn - Cn -1} is the group of cycles and where Bn = image{dn+ 1:Cn+ I -C,,} is the group of boundaries. The cellular chain complex above yields a functor

o (2.1.9) C* : CW/ -+ ChainZ Which determines the functor between homotopy categories C*:CW/= ChainZ/=. Foran abelian group A the homology of C. X ®A is the usual homology with coefficients in A, (2.1.10) H,,(X,A) =Hn(C*(X) ®A). 34 2 INVARIANTS OF HOMOTOPY TYPES

Similarly the cohomology of the cochain complex Hom(C* X, A) is the cohomology with coefficients in A,

(2.1.11) H"(X, A) = H"(Hom(C* X, A)). For A = 71 we also write H"(X, 1) = H"(X) and H"(X, 7L) = H"(X ). The Hurewicz homomorphism h is a natural homomorphism (n >- 1) h:wr"(X)-4H"(X), (2.1.12) h:7r"+t(Y,X) -->H,j(Y,X).

We define h by h(a) = a * e" where e"is an appropriate generator in H"( S") = Z or l-1" *,(CS", S") = 71 such that h is compatible with the exact sequences of the pair (Y, X); see (2.1.3). Now let X be a CW-complex with * E X°. We obtain, for n >_ 1, Whitehead's F -groups by the image group

(2.1.13) 1'"(X)=image(i*:a"X"-1 _lr"X") Hereis X" - 1 C X, is the inclusion of the (n - 1)-skeleton into the n- skeleton of X. Clearly r'"(X) = 0 for n = 1, 2 since it"(X" -1) = 0 in this case. Hence I'"(X) is an abelian group for all n. A cellular map F: X ---> Y induces a homomorphism f (F): I'"(X) - FM. The cellular approximation theorem shows that 1'"(F) depends only on the homotopy class of F so that we obtain a well-defined functor

(2.1.14) F,,: CW/= - Ab. Recall that a space X is n-connected if ir,(X) = 0 for i <_ n. The following lemma is well known.

(2.1.15) LemmaLet X be an n-connected CW-complex. Then there exists a homotopy equivalence Y - X where Y is a CW-complex with a trivial n-skeleton Y"=*.

The lemma implies for the 1-groups above the

(2.1.16) CorollaryLet X be an (n - 1)-connected CW-complex. Then 1;(X) _ 0 for i:5 n.

Now let X be a simply connected CW-complex. Then the Hurewicz map h = h" is embedded in the following long exact sequence which is the certain exact sequence of J.H.C. Whitehead, n >_ 2,

(2.1.17) ... ->H"+,X=-*FnX-'7r"X -_ H"X "I"X. The sequence is natural with respect to maps in CW/The operator i" is 2 INVARIANTS OF HOMOTOPY TYPES 35 induced by the inclusion X" CX. Moreover, the secondary boundary b" is defined by the following commutative diagram, where h is an isomorphism for n >- 3.

Hn(X",X"-1) h nTr(X",X"-') a > Trrt-1(Xn-1) U dog rn-1(x)

One readily checks that d h -' induces maps do and b" such that the diagram commutes. Let Z" be the set of n-cells of X E CW. Then

(2.1.18) C"(X) =71[Z"] is the free abelian group generated by the set Z". We obtain the basis Z" C C"(X) by the Hurewicz map

h: 7r"(X", X"-') o H"(Xn, X"- 1) = C"(X ) which carries the element ce E it"(X", X"-'), given by the characteristic map of the cell e, to the generator e E C"(X). We can choose a map

(2.1.19) f:A"= V S"-1 ->Xn-1 z" and a homotopy equivalence c: Cf = X" under X". Here A" is a one-point union of (n- 1)-spheres Se -' corresponding to n-cells e E Z" and Cf is the mapping cone Cf= CAn Uf Xn - '. We call f the attaching map of n-cells in X. The induced map

C"(X) =77[Z"] _ it"-1(An) ITn-1(X" coincides with the boundary map dh - 'in the diagram above. This yields a direct connection between the attachingmap f and the secondary boundary b0. The exactness of the sequence in (2.1.17) and Corollary 2.1.16 imply the following result.

(2.1.20) Hurewicz theoremLet X be an (n - 1)-connected CW-complex, n ? 2. Then h: 7r"(X) =H"(X)

is an isomorphism and h:ir" + 1(X) H" + 1(X) is surjective. 36 2 INVARIANTS OF HOMOTOPY TYPES

For an (n - 1)-connected space X we use the isomorphism h: 7rn(X) - Hn(X) as an identification. In addition to the Hurewicz theorem we have the following result due to J.H.C. Whitehead. For this we consider the functor

(2.1.21) r,: Ab - Ab given by 1';(A) = F(A) and f, (A) =A 0 71/2 for n > 3. Here r denotes Whitehead's quadratic functor and y: A -> f, (A) is the universal quadratic map for n = 2 and the quotient map for n >_ 3; see (1.2.1).

(2.1.22) TheoremLet n >t 2 and let X be an (n - 1)-connected CW-complex. Then there is a natural isomorphism :In'(HnX)F +1(X).

Hence the certain exact sequence yields the natural exact sequence Hn+2(X) b F'(H,, X)-I1iR+I(X)-aHn+l(X)-b0.

The homomorphism rr is induced by the Hopf map rin which is a generator ofTrn+ 1(Sn),n >: 2.Infact,foran elementa E HX representing a : Sn --X n CX the isomorphism i carriesya to the composite { a%}. Moreover the homomorphism 71 carries ya to alto. We shall use the exact sequence of Theorem 2.1.22 for the classification of (n - 1)-connected (n + 2)-dimensional homotopy types. One readily derives from the theorem the following isomorphisms for Eilenberg-Mac Lane spaces K(A, n) and Moore spaces M(A, n).

(2.1.23) CorollaryThere are natural isomorphisms

6: 7r3 M(A, 2) = F(A) = H, K(A, 2),

0: rrn+, M(A, n) =A ® 71/2 = Hn+2K(A, n),n>3.3.

Clearly H K(A, n) = A = -rrn M(A, n) and Hn+ iK(A, n) = 0 by the Hurewicz theorem. We also use the operators in Whitehead's exact sequence for the definition of the natural transformation (m > n)

(2.1.24) 0: irn M(A, n) --> Hm+ 1 K(A, n).

For this let k: M(A, n) -+ K(A, n) be the map which induces in homology the identity Hn(k) = 'A of A. Then 0 is the composite 0=b,-+IF,n(k)tmt:

7rmM(A,n)-1'mM(A,n) k'->1'mK(A, n)-H,n+IK(A,n). 2 INVARIANTS OF HOMOTOPY TYPES 37

Obstruction theory shows that the cohomology of X E CW can be described by the natural isomorphism (n >_ 1)

(2.1.25) Hn(X, A) = [X, K(A, n)], where [X, K(A, n)] is the set of homotopy classes in Top*/=. Let kA E Hn(K(A, n), A) = Hom(A, A) be given by the identity of A. Then the isomorphism carries the homotopy class 6 E [X, K(A, n)] to the cohomology f*kA. We also have the natural isomorphism

(2.1.26) H"(X, A) = [C* X, C* M(A, n)].

Here the right-hand side is the set of homotopy classes in Chain,/=. Dually we define the pseudo-homology

(2.1.27) Hn(A; X) = [C*M(A,n),C*X] with coefficients in A, not to be confused with Hn(X, A) above. One has the following homotopy classification of chain maps; see for example 10.13 in Dold [AT].

(2.1.28) TheoremLet C and D be chain complexes in Chain, and assume Cn is free abelian for all n. Then there is a natural short exact sequence

Ext(H*_1C,H*D)N[C,D] -»Hom(H*C,H*D)_ of abelian groups. Here [C, D] is the set of homotopy classes of chain maps and Hom(H* C, H* D) is the group of degree 0 homomorphisms H* C -- H* D, that isthe product of all groups Hom(H,,C, HD) for n E Z. Moreover Ext(H*_1C, H* D) is the product of all groups Ext(H,_1C, H, ,D) for n E Z. The map µ carries a chain map F: C - D to the induced homomorphism µ(F) = F*, The exact sequence is split(unnaturally).

As a special case of the theorem we get, for the cohomology Hn(X, A) and pseudo-homology H,(A, X), the following universal coefficient formulas.

(2.1.29) CorollaryFor X E CW and n >_ 1there are natural short exact sequences

Ext(H,_ 1X, A) N Hn(X, A) Hom(H,, X, A)

Ext(A,Hrt+1X) ° Hom(A,H,,X).

The sequencesare split (unnaturally). 38 2 INVARIANTS OF HOMOTOPY TYPES

There are the following two ways of representing elements in the group Ext(A, B) where A and B are abelian groups. On the one hand each short exact sequence B E-.A of abelian groups represents an element {E} E Ext(A, B). On the other hand, we use a free resolution of A; this is a short exact sequence

A, -*d Ao -A where A0, A, are free abelian groups. Then d induces a homomorphism d* = Hom(d,1): Hom(A0, B) --> Hom(A,, B) and we may define Ext(A, B) to be the cokernel of d*. Thus homomorphisms g1 E Hom(A1, B) represents elements {g1) E Ext(A, B). We have the well- known

(2.1.30) Lemma The elements {E} and {g1} in Ext(A, B) above coincide, that is {E} = {g1} if and only if there is a commutative diagram

A, >-* A0 -A

II 191 I go B >-+E -'A

We shall also use the following facts about chain complexes; for a more detailed discussion we refer the reader to XII §4 in G.W. Whitehead [EH]. Let C be a chain complex of free abelian groups. Then the universal coefficient theorem asserts that there is a short exact sequence (see Theorem 2.1.28)

(2.1.31) 1C, A)%H"(C, A) - A) which is natural with respect to both chain maps F: C -* C' and homomorphisms f: A --+A'. The sequence admits a splitting (unnaturally). Let u E H"(C, A). If a: C -A is a cocycle representing u, then the restriction a I Z,,: Z, -->A maps the group B of bounding cycles into zero and thereby induces a homomorphism u * = µ(u): H (C) -+A (1)

with "*(x)=a(x) for x E Z,,. This defines µ in (2.1.31). Now lete EE Ext(H_,C, A). The group H,,_ 1C has the convenient free resolution B, I-+ Z, I -" H, 1C. (2) 2 INVARIANTS OF HOMOTOPY TYPES 39

Let b: B"_ 1-A be a homomorphism representing e, see Lemma 2.1.30. Let d: C,, -. B,,_ 1be defined by the boundary in C. Then bd: C,, -A is a cocycle, representing the element 0(e) = {bd}. This defines A in (2.1.31). For u E H"(C, A) we have by (1) the exact sequence q H"(C) - A cok u * - 0 (3) where q is the quotient map. Clearly qu * = 0. In view of the naturality of the exact sequence (2.1.31) there is a commutative diagram

Ext(H"_ 1C, A) »H"(C, A) A Hom(H"C, A) 14. 1q. q. 11 A µ Ext(H"_ 1C, cok u *) N H"(C, cok u *) - . Hom(H" C, cok u * ) Since q * µ(u) = q* u * = qu * = 0 we see that the element

U (4) is well defined. Thus each element u E H"(C, A) determines canonically a pair of elements (u*,ut). The operations u - u *, u H ut have naturality properties which are explained in G.W. Whitehead [EH]. We need the following property of the element ut with respect to mapping cones in the category of chain complexes. Let f: C --> D be a chain map between chain complexes of free abelian groups. The mapping cone of f is the free chain complex Cf with (C1)"=C"-1 ®D" (2.1.32) d(x, y) = (dx, f(x) -dy). We obtain the short exact sequence of chain complexes (cofibre sequence)

ir D> Cf -sC. (1) Here i is the obvious inclusion and sC is the suspension of C, that is the mapping cone of C -> 0. Then ar is defined by ir(x, y) = x. We clearly have H"sC = H"_ 1C. The short exact sequence (1) yields the exact homology sequence HC_LH"D-.H"Cf-H"_1C H- if H"_1 ,, D (2) which in turn gives rise, for each n, to a short exact sequence

cok(H,, f) H H"C f -. ker(H"_ 1f) (3) representing an element {H"Cf}EExt(kerH"_1f,cokH"f). (4) 40 2 INVARIANTS OF HOMOTOPY TYPES

We proceed to explain how the operator u H ut above leads to a description of this extension element. A cohomology class u E H"(D, H D) is said to be unitary if and only if the homomorphism u* is the identity of H"D. Let u be such a unitary class and consider the element f *u E H"(C, H D). Then (f*u)* =H"f so that (f*u)t EExt(H"-1C,cokH"f).

For the inclusion j: ker(H"_ 1f) c H, -1C inducing

j*: Ext(H"_,C,cok H" f) Ext(kerH"-1 f,cok H" f ) we get the equation:

(2.1.33) Theorem (H"CfI = -j*(f *u)t.

For a proof see XII.4.9 in G.W. Whitehead [EH]. We point out that the theorem can be applied to topological mapping cones. In fact let f : X -* Y be a cellular map. Then the mapping cone Cf is a CW-complex for which C* (Cf) is the mapping cone of the chain map C* F: C* X --> C* Y.

2.2 F-Groups with coefficients

Using a Moore space M(A, n) we obtain the homotopy groups of a pointed CW-complex X with coefficients in A by the group of homotopy classes (n y 2) it"(A, X) = [M(A, n), X ]. Here M(A, n) _ V"-1 MA is an (n - 1)-fold suspension. For the pseudo-homology H"(A, x) _ C. M(A, n), C,, X ] we thus have the Hurewicz homomorphism

(2.2.1) hA: a"(A, X) --> H"(A, X) which carries the homotopy class of a cellular map x: M(A, n) -* X to the homotopy class of the induced chain map C. W: C* M(A, n) - C,, X. For A = Z this Hurewicz homomorphism coincides with the classical homorphism in (2.1.12) above. Moreover the universal coefficient sequences yield the commutative diagram

Ext(A,irn+1X) H it"(A,X) : Hom(A,ir"X)

(2.2.2)

Ext(A,H"+,X) >'I H,(A,X) Hom(A,H"X)

The classical Hurewicz homomorphism h = hz is embedded in Whitehead's 2 INVARIANTS OF HOMOTOPY TYPES 41 exact sequence. We want to show that also the homomorphism hA is part of such an exact sequence. For this we introduce new `F-groups with coefficients'. As the classical F-groups F"(X) of J.H.C. Whitehead these new groups are derived from the homotopy groups of the skeleta of a CW- complex X. Recall that r ,,(X) is defined by the image

F" = r"X= image{i,, : ar"X"-' -> X"} where is X"-' cX" is the inclusion.

(2.23) DefinitionLet A be an abelian group and let X be a 1-connected CW-complex. We have the canonical inclusion and projection respectively

1: F"X>-a Tr"X", (1) p: 7r"+iX" -. F +,X.

We use these for the definition of the groups F"(A; X) for n >_ 3 as follows. Consider the commutative diagram

Ext(A, F"+ 1X) Ext(A, arrt+ 1X") P. push I F"(A;X) P >--a ar"(A; X") (2)

Hom(A,F,,X) >-' Hom(A,ir"X")

Here `pull' and `push' denote a pull-back diagram and a push-out diagram respectively in the category of abelian groups. Thus the F -groups with coefficients f,,(A; X) are embedded in the short exact sequence

Ext(A,F'n+1X)NF,, (A; X) -µ* Hom(A,F"X). (3)

Clearly, thissequence is natural with respect to cellular maps X -> Y since, for the restriction X"--> Y" of such a map, all arrows are natural. Below we show that cellular maps X -p Y which are homotopic induce the same homomorphism r"(A; X)- r"(A;Y). This shows that (3) is actually a homotopy invariant of X. The group F"(A; X) is an abelian group for all n E Z. For n < 1 we set F"(A; X)= 0 and we set

F2(A;X)=Ext(A,F3X) (4) 42 2 INVARIANTS OF HOMOTOPY TYPES since r2 X = 0. The group I'"(A; X) generalizes the r-group of J.H.C. Whitehead since we clearly have [,,(Z; X) = Hom(Z, I'" X) = r,, X. If X is a pointed CW-complex which is not simply connected we set

T"(A,X)=r,,(A,X) (5) where X is the universal covering of X with a base point * mapping to the base point of X. If X is just a pointed space we define I'"(A,X)=r(A,ISX) (6) where ISX I is the realization of the singular set of X. In this book, however, we consider mainly 1-connected CW-complexes.

(2.2.4) Remark For an n-dimensional CW-complex X" with ir1X" = 0 we have Whitehead's exact sequence

F" r._ = r" X" N i" X"h# H" X" -' 1 where Z,, =H"X" is free. Therefore im(h) is free and thus i:I'" - 7r,, X' admits a retraction. This shows that i * in (2) above is injective.

(2.2.5) PropositionLet F, G: X - * Y be cellular maps between simply connected CW-complexes. If F and G are homotopic we have F* = G* : F, (A; X) ---* F,(A;Y).

This shows that f'"(A;.) is a well-defined homotopy functor on the category of simply connected spaces or more generally on Top*/=.

Proof of Proposition 2.2.5Let y: M(A, n) -- X" be a map with µ(y) E im(i*). Then we know that the restriction of y to the n-skeleton of M(A, n) actually factors up to homotopy over X"'. Thus we can assume that y is a pair map

y: (M(A, n), M(A, n)") -* (X", X"- '). (1) Since X is simply connected this map is a twisted map between mapping cones; compare V.7.8 in Baues [AH]. We point out that y shifts the dimension; the n-skeleton of M(A, n) is mapped to the (n - 1)-skeleton of X. Next consider the restrictions F", G": X" Y" of the cellular maps F and G. Since F = G there is

a : M(C" X, n) --> M(C"+ 1Y, n) = M (2) with G"=F"+(gn+1)*(a)in[X",Y"]. (3) 2 INVARIANTS OF HOMOTOPY TYPES 43

Here C,, X, C*Y are the cellular chain complexes and gn+ 1: M -+ Y" is the attaching map of (n + 1)-cells in Y. By (3) we get in it (A,Y") the equation G*{y) =y*(F" +gn+1{a)) (4)

=y*F" + Dy (gn+1{a}, F") (5) where Vy is Et: with 6 associated with the twisted map y; see V.3.12 in Baues [AH]. In (5) the addition is given by the mapping cone structure of M(A, n). Therefore /3 = {oy (gn+ 1{a}, F")} E Ext(A, irn+1Yn) and (5) is equivalent to G*{y}=F"(y)+0(/3) (6) with 0 defined by the universal coefficient sequence; here + is the group operation in 7rn(A, Y"). We claim that for p: r rn +1Y -* 1 7 ,1 Y we have p*/3=0inExt(A,rn+1Y) (7) This shows that the term S vanishes in QA; Y); see (2) in Definition 2.2.3. Thus Proposition 2.2.5 is a consequence of (7) and (6). We check (7) as follows. We have the commutative diagram

77n+1(Mv Yn)2 n+1(Yn) Tin+1(Y"-1)

I rn+ 1Y where i* (gn+1)* = 0 since ign+ 1 = 0. This shows that p(gn + 1,1) * = 0. Since by definition of /3 above /3 E image Ext(A, (gn+1, 1) we get p * /3 = 0. This completes the proof of Proposition 2.2.5. The I'-group rn(A, X) is natural in A in the following sense. Let cp: A -> B be a homomorphism between abelian groups and let fp: M(A, n) --> M(B, n) be a map between Moore spaces which induces1P in homology. Then iP induces a homomorphism (2.2.6) fP*:I'n(B, X) -* rn(A, X) since all arrows in Definition 2.2.3 are natural with respect to gyp. In particular the following diagram commutes

Ext(B,I'n+1X) rn(B,X)-",Hom(B,r,,X) (2.2.7) W. I ;V I ,F. Ext(A,rn+1X) - rn(A,X) -.Hom(jA,r"X) 44 2 INVARIANTS OF HOMOTOPY TYPES

For a E Ext(A, B ® Z/2) and n >_ 3 we obtain a map 4i + i(a): M(A, n) -- M(B, n) which induces cp in homology; here we identify B ® Z/2= rr, 1 M(B, n). Now we get the formula

(2.2.8) (;p+M(a))* = ip* + Act *A where a *: Hom(B, r" X) --> Ext(A, r,,, X) is defined by

a*(y) = (77(y ® 71/2))* (a).

Here the homomorphism 77: r"(X) ® Z/2 - F, (X) is induced by the Hopf map q,,: Sn+1 -I S", thatis,for 6: S" -*X"-' with E r"(X) we set 1) = (i677") where is X"-' cX" is the inclusion. The formula for (i-p + A(a))* above is obtained in the same way as in (1.3.13).

(2.2.9) Definition We here define two natural subgroups F,, (A, X) and F (A, X) of the group I'"(A, X). For n >_ 2 let

F, (X) = kernel(i: F ,,(X) --> rn+1(X)) where rl is induced by the Hopf map. Moreover let

r;;(X)=image(b"+ X: H"+1X--*F"X) = kernel (.X: r"X--+ it"X).

In Lemma 2.3.4 below we show that we have the natural inclusions r"(X) c rF(X) c r"(X).

These groups are trivial for n = 2. We obtain the corresponding binatural inclusions F, (A,X)cF,(A,X)cF,(A,X)

by the following pull-back diagrams in which the rows are short exact.

Ext(A,r"+1X) N r"(A,X) -A Hom(A,r"X)

Ext(A,rn±1X) N F,,(A,X) -. Hom(A,F,,X)

Ext(A,r"+,X) N r;;(A,X) - Hom(A,r;;X) 2 INVARIANTS OF HOMOTOPY TYPES 45

All vertical arrows are inclusions. The advantage of I' is that a homomorphism cp: A ---b B induces a homomorphism cp* = iP*: f,(B, X) - f,(A, X), which by (2.2.8) above does not depend on the choice of irp; compare the definition of7r, ,(A, X) in Definition 1.3.14. Hence we get well-defined bifunctors, n >: 2,

r,',, I;; : Ab°P x Top*/= -9 Ab together with the natural short exact (A, µ)-sequences above. Clearly if A = Z we have I,,(7L, X) = Fn(X) and I;,'(l, X) = F, (X). Below we shall see that f, (A, X) is naturally isomorphic to a pseudo-homology group; see Theorem 2.6.14(4).

2.3 An exact sequence for the Hurewicz homomorphism with coefficients

We generalize Whitehead's certain exact sequence by introducing coefficients in abelian groups. We describe the operators of the sequence explicitly in terms of the CW-structure of a CW-complex X; in the next section we give an alternative construction by a fibre sequence. Let X be a simply connected CW-complex and let A be an abelian group. Then there is the long exact sequence (x E Z)

(2.3.1) b.,I --->H"+1(A,X) f,,(A,X)=__0

Here h = It' is the Hurewicz map for homotopy groups with coefficients in A, see (2.2.1), and i =iR is induced by the inclusion X" cX, see Definition 2.2.3. The boundary operator b" = is explicitly constructed in Definition 2.3.5 below. The universal coefficient sequences yield the commutative diagram (2.3.2) Ext(A,Hn+2X)b- Ext(A,I'"+cX)=Ext(A,ar".,X)*Ext(A,H"+,X)

Hn+1(A,X) bI'n(A,X) = r"(A,X) H"(A,X)

Hom(A,Hn+1X) b'- Hom(A,I'"X) Hom(A,a"X) Hom(A,H"X) 46 2 INVARIANTS OF HOMOTOPY TYPES

Here the top row and the bottom row are induced by the classical certain exact sequence of J.H.C. Whitehead.

(233) TheoremThe sequence (2.3.1) is exact and natural with respect to maps ip: M(A, n) -> M(B, n) and X - Yin Top*/-. Moreover diagram (2.3.2) commutes.

Clearly for A = Z the exact sequence (2.3.1) coincides with Whitehead's exact sequence. For the definition of the boundary operator b,4 we need the (2.3.4) Lemma Let7: F,,- I(X) --> fn(X) be induced by the Hopf map 77" _ 1: S" -i S- 1Then the composition

0 = 77b": H"(X) - r,,-1(X) -+ F,,(X) is trivial.

Proof We consider the following commutative diagram

HnX-b-4 1X H,, X' -. F, rnX n n n

lrn 1X nI n-1Xrt Cn`Y f - _ rlnt 7TnXn where jfn = 0 and therefore qb" = 0. 0

(2.3.5) DefinitionFor a 1-connected CW-complex X we define the boundary operator

as follows, n ;-- 3. We may assume that X 1 = *. Let C = C * X be the cellular chain complex of X and let Z" = ker d" and Bn = im dn+ 1 be the groups of cycles and of boundaries in C respectively. Then

d B. - Z. -' Hn (1)

is a free presentation of H" and we can choose the Moore space M(H", n - 1) to be the mapping cone of

d:M(Bn,n-1)--*M(Z",n-1). (2)

Here d is given by (1) and by the canonical bijection

[ M(G, n), X ] = Hom(G, ir,, X) (3) 2 INVARIANTS OF HOMOTOPY TYPES 47 which exists whenever G is a free abelian group. We choose for the exact sequence

Z" N Cn Bnt (4) a splitting t so that Cn = Zrt ® tBn -. Let

f, - ITrtXn (5) be the attaching map of (n + 1)-cells in X. Since the diagram

Cn+1 f . 1T"Xn U U Zn+ Hn + 1 b rn P (6) XII 1rn commutes (where b is a lift of brt+ 1 p) we can assume that the attaching map fn+, yields the commutative diagram M(Cn+l,n) - Xn fn*I U U (7) M(Zn+1,n)bX"-1 Here we have by (4)

M(Cn+1,n) =M(Zrt+t,n) V M(tB,,,n). (8)

A crucial step for the construction of b' is the `principal reduction' of the CW-complex X as described in (3.5.13) of Baues [OT], compare also VII 3.3 in Baues [AH]. The principal reduction yields a map

v:V-*X'-1(n>:3) (9) with the following properties (10) and (11): V is a CW-complex with cells in dimension n- 1 and n together with a cellular homotopy equivalence

IV=: Xrt+ 11Xn-1 (10) for the suspension IV. There is a cellular homotopy equivalence

Ct _ Xn+1 (11) 48 2 INVARIANTS OF HOMOTOPY TYPES under Xn-' which induces the identityon C,* Xn+'; Ct, denotes the mapping cone of v in (9). Now (2), (4), and (8) imply V=M(Zn+1,n)VM(Hn,n-1)VM(tBn_1,n-1) (12) SlV"-1=M(Cn,n-1)=M(Z,,,n-1)VM(tBn_1,n-1).

Here the inclusion j: V"-' c V yields the identity on M(tBn_1, n -1) and yields the canonical inclusion M(Zn, n - 1) c Cd = M(Hn, n - 1) given by (2). The restriction of the map v in (9) to the subspaces of V in (12) have the following properties:

VIM(z., ,n)=b, (13) see (7), and

VI ,n-I)= ,n-1)' (14)

Moreover, for 6 = v I M(H,,,n - 1)the diagram M(Hn,n-1) 0X"-'

U U 15) M(Zn, n - 1)bX"-2 commutes. This shows that the element 13 represents an element

{p}El'n_1(Hn,X)withµ{$}=bn. (16) We use the element (p) for the definition of the boundary operator bn above. Let {a) E H,,(A, X), that is

achain map. Using a splitting C,,,, = Zn+1® tBn as in (4) we obtain from a the commutative diagram

Cn+1M(A,n) Zn+I®tBn=Cn+)

1(°,d) Id I (18) CnM(A,n) Q Zn C Cn

Here a,, 1 has the coordinates a,,1 =(0/r°, a') where a' is the restriction of an and where 1/, ° represents an element 4iQ E Ext(A, Hn+ I). The chain map a 2 INVARIANTS OF HOMOTOPY TYPES 49 induces a * = cpa E Hom(A, H,,)in homology. We choose a realization lpa: M(A, n) - M(H,,, n) of rpa which induces ;aa:rn-1(H,,,X)->I'rt_(A,X). Now we define the boundary operator bn above by the formula bn(a)=cpQ{ }+0(bn+1)(20) Here we use { ) in (16) and (b,,1) * : Ext(A, H,, Ext(A, r,,). The element (20) is well defined. In fact, for a different choice 4a + a which realizes the homomorphism gyp, we have by (2.2.8) (21) where a *µ{ /3 } = a *bn = (i(bn ® Z/2)) *(a) with rtb,, = 0 by Lemma 2.3.4. Similarly we see that b' (a) depends only on the homotopy class (a) of the chain map a.

(2.3.6) Lemma The boundary operator bA is natural in X. Proof Let F: X - Y be a cellular map between CW-complexes with X1=*=Y1.Let

w:W-Yrt-1 be chosen for Y (with C. = Y"") in the same way as v in Definition 2.3.5 is chosen for X. The cellular map F yields the map

F:(C,,,Xn-1)-,(CW,Y"-1) between mapping cones which, by V.3.12 in Baues [AH], is a twisted map since n z 3. Therefore, there is a homotopy commutative diagram

-a WVYn-1 (1) 6o

1(w, 1) 1, Xn-1 Yn-1 -00 which is associated with F, see V.3.12 in Baues [AH]. Here 710is the restriction of the cellular map F and fois a map with the following properties. Let q:M(H,,,n-1)=Cd- M(Bn,n) 50 2 INVARIANTS OF HOMOTOPY TYPES be the quotient map. Clearly, q* induces the inclusion 0 in the universal coefficient sequence. Now the restriction

f= = S01M(H,,,n-1) (2) is the sum of the following four maps: 61:M(H",n - 1) -q M(B",n) ->M(Z;,+l,n) cW 2:M(H",n-1)--M(H,,n-1)cW

63:M(H",n-1) *M(B",n)->M(Bn_1,n-1)cW Y2.

Here 63 factors over q since V -* W V Y'- ' - W induces the chain map C. F on cellular chains. Moreover, , factors over q for dimensional reasons, since 4 is trivial on M(H, n - 1) and onY"-1; also 4 factors over q and maps to the subspace M(C, n - 1) V Y2 C W V Y"-' for dimensional reasons (compare the Hilton-Milnor theorem). The map lpn: H" -, H,,is induced by F in homology, cpn = Hn(F). Since X" is the mapping cone of fn we know that for is Xn-' cX" the composition if,, = 0 is trivial. Therefore also the projection p with

7Tny"-'-* FFY P n n

satisfies p(fn) * = 0. This shows that (w,1) * 63 and (w,1) * 64vanish in I'_ ,(H,,X, Y ); see Definition 2.3.5(2). Moreover 6, represents an element {61) E Ext(H,,X, H,1Y) such that A(bn+1)*{61) E rn-1(H"X,Y) (3) is represented by (w,1) * 6',; here we use Definition 2.3.5(6) for Y. By definition of the boundary bn we now have, on the one hand, (see Definition 2.3.5(20)) F*bR (a) =;paF*{/3}+A(bn+1)*F*(4/o). (4) On the other hand, we get for b = (C * F)a, resp. {b) = F* (a),

b,'{b}=;Pb{l3'}+A(bn+1)*+#b (5) Here 0' and 1(ib are chosen for Y as in Definition 2.3.5 and Pb = 4Pn gyp: A -. H" X H"Y. Now we have, by the commutativity of (1) and by (2), (3),

F*( p) = ip,*(J3') +0(bn+1)*R1}. (6) 2 INVARIANTS OF HOMOTOPY TYPES 51

Thus (4) = (5) since we get

F*(yia)+(pQ(61}_/b (7) by definition of tlia, ilib and S1; see Definition 2.3.5(18).

Proof of Theorem 2.3.3The naturality of the sequence (2.3.1) with respect to maps X -* Y is obtained by Lemma 2.3.6 since one readily checks that also iA and hA are natural with respect to such maps. Moreover it is easy to derive the naturality of (2.3.1) with respect to maps ip: M(A, n) ---> M(B, n) from the definition of the operators b A, i A, h A. Also the definitions show readily that bAhA = 0, hAiA = 0, and iAbA = 0 and that diagram (2.3.2) commutes. Thus it remains to check exactness. Since we give an alternative proof in the next section we leave this to the reader.

2.4 Infinite symmetric products and Kan loop groups In this section we compare the T-groups of J.H.C. Whitehead with the homotopy groups of a certain space FX. This is related to results of Dold and Thom on the infinite symmetric product of X and to results of Kan on the loop group of X. The results here are useful background knowledge on the r-groups and on Whitehead's exact sequence. In our proofs, however, we will always use the more direct definition of the I'-groups in terms of the skeleta of a CW-complex; see Sections 2.1 and 2.2. In particular our explicit construction of the secondary boundary b,A in Definition 2.3.5 is an essential step for the proof of the boundary classification theorem below; a definition of b' as given here by a fibre sequence is not appropriate for this proof. Let X be a simply connected CW-complex with base point and let SPX be the infinite symmetric product of X. This is the limit of the inclusion maps

(2.4.1) X=SP'XcSP2XcSP3Xc .

Here SP"X = (X x ... X X)/S(n) is the quotient space of the n-fold product X x ... X X by the action of the symmetric group S(n) which permutes the coordinates. By the result of Dold and Thom [SP] we have the natural isomorphism

(2.4.2)

His the integral homology of X. Moreover, the inclusion X - SPX induces the Hurewicz homomorphism

(2.4.3) h:1rX-+zrSP=X=H(X). 52 2 INVARIANTS OF HOMOTOPY TYPES

Let rX be the homotopy theoretic fibre of the inclusion is X c SPX. Then the fibre sequence (2.4.4) rX- X--SP'X yields an exact sequence of homotopy groups which by (2.4.3) has the following form h-* Hn+1X-- -- This sequence is similar to Whitehead's exact sequence in Section 2.1. In fact, since X is 1-connected we have a natural isomorphism (2.4.5) -rXrnx such that the diagram H,,+IX- 'rnrX-+ (2.4.6) it II I II -H,,+IX_' f,X -irX_'HHX_* ... commutes. Here the bottom row is Whitehead's exact sequence and the top row is induced by (2.4.4) and (2.4.3). A different approach is due to Kan [CW]. This result of Kan can be used for the proof of (2.4.5) and (2.4.6). Let Y be a reduced simplicial set, for example let Y = SX be the reduced singular set of the space X. Then Kan defines the loop group GY which is a free simplicial group. The realization IGYI is a topological group which is equivalent to the loop space [IIYI- For F = GY denote by [F, F] C F the commutator subgroup, i.e. the simplicial subgroup such that [ F, F In = [ F,,, F, ] for all n. Then Kan proves that there is a natural equivalence (2.4.7) 1[GY,GY]= rnIYI if IYI is simply connected. Now consider the fibre sequence (2.4.8) [GY,GY] with AY = GY/[GY, GY]. Here AY is the simplicial group for which (AY)n is the group (GY)n made abelian. The map i is the inclusion and p denotes the projection. Kan proves that there is a natural equivalence 1r_ 1AY = and that p induces the Hurewicz homomorphism. Therefore the homotopy sequence of the fibre sequence (2.4.8) is the top row in the following commutative diagram

Hi + 1Y - vri_ 1[ GY, GY ] -'Trn _ IGY -+ HnY (2.4.9) II II II II Hn+ IIYI FnIYI -1rnjYI - HlYI. 2 INVARIANTS OF HOMOTOPY TYPES 53

The bottom row is Whitehead's exact sequence. Commutativity of (2.4.9) was proved by Kan. Now (2.4.7) yields (2.4.6) since there is the equivalence of fibre sequences: I[GY,GY]I -IGYI - IAYI

(2.4.10) =1 =1 =1 flfX-> flX -> fiSP'°X. The vertical rows in the commutative diagram are homotopy equivalences (X is simply connected and Y = SX is the reduced singular set of X as above). Next we consider homotopy groups of the space FX with coefficients in an abelian group A. The natural equation ir rX = r X leads to the short exact coefficient sequence (see (2.2.2))

(2.4.11) Ext(A,ri+1X)N which by a construction similar to the one of Kan shows that we have a natural isomorphism

(2.4.12) 1r (A, rX) - r (A, X). Here the right-hand side is the IF-group with coefficients constructed in Definition 2.2.3; the isomorphism is compatible with 0 and it in (4.1.11) and Definition 2.2.3(3) respectively. For A = Z this is just the same isomorphism as in (2.4.5). For our purposes the definition of r (A, X) by the skeleta of X is more appropriate. On the other hand we have the homotopy groups of the spaces SP'X with coefficients in an abelian group A. The result of Dold and Thom (2.4.2) yields the short exact coefficient sequence

(2.4.13) as in (2.2.2) which, by the equivalence IAYI11SPX in (2.4.10), shows that we have a natural isomorphism (2.4.14) ir,,(A, SPX) -

H (A, X) X ]is the pseudo-homology defined by homotopy classes of chain maps; see (2.1.27). The isomorphism (2.4.14) is compatible with A and µ in (2.4.13) and Corollary 2.1.29 respectively. For the proof of (2.4.14) we also use the Dold-Kan theorem (see for example Dold and Puppe [HN]) which shows that the simplicial abelian group AY in (2.4.10) is completely determined by the singular chain complex C. IYI which is equivalent to the chain complex C. X. The pseudo-homology groups 54 2 INVARIANTS OF HOMOTOPY TYPES

SP=X) were also considered by Hilton [HT] (Chapter 5) who showed that (2.4.13) is always split. We here point out that these groups can easily be described by chain maps as in (2.4.14). The fibre sequence (2.4.4) yields for homotopy groups with coefficients in A an exact sequence which, by (2.4.12) and (2.4.14), has the following form

(2.4.15) irn+l(A,SP-X) it" (A , X) ---p 7r" (A, SPSX )

II II H"+(A,X) 6 > T'(A,X) ,,,,,(A, X) h >H"(A,X)

Here the top row is the usual fibre sequence given by (2.4.4) and the bottom row is our exact sequence constructed in (2.3.1). Using the construction of the isomorphisms (2.4.12) and (2.4.14) one can check that the diagram commutes; see for example (2.4.3) and (2.4.8).

2.5 Postnikov invariants of a homotopy type

We describe in this section the classical `k-invariants' which were introduced by Postnikov in 1951; they were also studied by J.H.C. Whitehead [GD]. We show that Postnikov's invariants of a homotopy type have properties which are exactly analogous to the properties of the boundary invariants in Section 2.6 below. As pointed out by J.H.C. Whitehead [I] one has to consider the hierarchy of categories and functors

(2.5.1) 1-types P- 2-types . P- 3-types where n-types are connected CW-spaces Y with zriY = 0 for i > n and where n-types is the full subcategory of Top*/= consisting of n-types. The functor P which carries (n + 1)-types to n-types is given by the Postnikov functor

(2.5.2) CW/= -* n-types. Here CW/= is the full homotopy category of CW-complexes X with X ° = *. For X in CW we obtain P,, X by `killing homotopy groups'; that is, we choose a CW-complex P,,X with (n + 1)-skeleton

(P"X)"`' =X"+'and ir;(P"X)=0fori>n.

For a cellular map F: X -- Y in CW we choose a map PF"+': P"X P"Y which extends the restriction F"+': X"+' --- y"+i of F. This is possible by usual arguments of obstruction theory since zr;(P" X) = 0 fori > n. The 2 INVARIANTS OF HOMOTOPY TYPES 55 functor P" in (2.5.2) carries X to P" X and carries F to the homotopy class of PF"+ tDifferent choices for P" X yield canonically isomorphic functors. Since 1-types are the same as Eilenberg-Mac Lane spaces K(ir, 1) we can identify a 1-type with an abstract group. In fact, let Gr be the category of groups. Then the fundamental group 7r, gives us an equivalence of categories

(2.5.3) 7r,:1-types --> Gr together with a natural isomorphism7r 1(P, X) = 7r,(X) where P, X = K(7r1X,1). From this point of view n-types are natural objects of higher complexity extending abstract groups. Following up this idea Whitehead looked for a purely algebraic equivalent of an n-type, n > 2. An important requirement for such an algebraic system is realizability, in three senses. In the first instance this means that there is an n-type which is in the appropriate relation to a given one of these algebraic systems, just as there is a 1-type whose fundamental group is isomorphic to a given group. The second kind is the `realizability' of `homomorphisms' between such algebraic systems by maps of the corresponding n-types. The third kind is a 1-1 correspondence of such homomorphisms and the homotopy classes of maps between n-types. For example the functor 7r, in (2.5.3), which carries a 1-type to its fundamental group, satisfies these three properties. We have further examples for certain subcategories of the category of n-types. Let

(2.5.4) typeset m be the full subcategory of n-types consisting of (m - 1)-connected n-types. Then we have for m > 2 the equivalence of categories

(2.5.5) 7rm : types' -L- Ab where Ab is the category of abelian groups. This equivalence is the higher- dimensional analogue of the equivalence (2.5.3). An object in the category types',thatisan (m - 1)-connectedm-type,isthe same asan Eilenberg-Mac Lane space K(A, m) which is determined by an abelian group A.

(2.5.6) Definition A map

p": X->P"X (1)

which extends the inclusion X n + 1 c P" X in (5.2) is called the n-typeor a Postnikov section of X. Clearlyp" induces isomorphisms of homotopy groups

(p"),,:1r;X-7r;P"Xfori

The map pn is natural with respect to the functor in (2.5.2). The Postnikov tower {qn} is given by maps (3) with qn pn = pn_ 1.Such maps are well defined up to homotopy. For further properties of the Postnikov tower we refer the reader to Baues [OT] and Baues [AH]; compare also G.W. Whitehead [EH], Spanier [AT], Mosher and Tangora [CO], and many other books on homotopy theory.

(2.5.7) DefinitionLet X be a simply connected CW-complex and let A be an abelian group. We define the abelian group

$n_1(X, A)= Hn+1(Pn_1X, A) (1) by use of the Postnikov functor 1, see (2.5.2), and by the cohomology with coefficients in A. Thus 13,,-, is a bifunctor $n_1:spaces'xAb-*Ab (2) where spaceS2 is the full homotopy category of simply connected CW-spaces. The (n - 1)-Postnikov section pn- 1:X - + P- 1X induces the following commutative diagram for Whitehead's exact sequence (see also 11.4.8 in Baues [CH]) HnPn-1X = rnpn-1X ' 0 H,,P.-1X"rn-1Pn-1X

(3)

Hn + 1X -brn X - irr X - HH X --+ r.-,X

Here the vertical arrows are induced by p,,-,. Thus we have natural isomorphisms ® = (pn_ 1)*1b 6:HHPn_1X=F,_IX ©: Hn+1Pn-1X =rFX (4) where r,,'- 1 X = ker(in _ 1X) is the image of the operator bn X : H X -+ r,,_ 1X. We use these isomorphisms as identification. Thus the universal coefficient formula for the cohomology group (1) leads to the commutative diagram of short exact sequences

Ext(HnPn_1X,A)>%* Hn+1(Pn-1X,A) -Hom(Hn+IPn-1X,A)

II II II (5) Ext(rn_1X,A) 1(X,A) A Hom(rnX,A) 2 INVARIANTS OF HOMOTOPY TYPES 57

The diagram is natural in X and A. Given an element k E 3n _1(X, A) we obtain elements (see (1.2.31))

k* = µk E Hom(FnX, A) kt = A-'q* (k) E Ext(r,;_,X,cok k* ). (6)

Here q: A --p cok k * is the projection for the cokernel of k * and

q.: I(X, A) - $n-,(X,cok k*) (7) is given by the functor in (1). Since clearly µq * (k) = q,, µk = q k * = qk * = 0 we see that kt in (6) is well defined. We use the for the following definition of Postnikov invariants of X.

(2.5.8) DefinitionLet X be a simply connected CW-complex. Then we obtain the Postnikov invariant or k-invariant (n ;-> 3)

k =knXE $n_1(X,ir,,X) (1) as follows. For A = Tr X we have the fibre sequence

K(A,n)i.--* (2) where qn is defined by the Postnikov tower of X, see Definition 2.5.6(3). It is clear that the fibre of qn is an Eilenberg-Mac Lane space K(A, n). We thus obtain the classifying map k in (2) which represents k in (1) by the transgression element

kn = (q*) -'dA- 1 (1,,X) (3) where E Hom(A, A) is the identity of A = Trn X. Here we use the homomorphisms

Hom(A,A)H"(K(A ,n),A) Hn+1(Pn 1X, A) vl _ (4) Hn+'(P,,_1X, *,A) where d is the boundary for the pair P,, X, K(A, n) given by jn in (2). The map q*, induced by qn in (2), is an isomorphism since X is simply connected. Compare (5.2.9) and (5.3.2) in Baues [OT]. Using the notation in Definition 2.5.7(6)we obtain by knX the homomorphism

X (5) 58 2 INVARIANTS OF HOMOTOPY TYPES which, as we shall see in Theorem 2.5.10, coincides with the operator i,,X in Whitehead's exact sequence. We thus get by Definition 2.5.7(6) the element

(k"X)tE Ext(f, _,X,coki"X) (6) where r," X = ker(i"_ ,X). On the other hand, the exact sequence I',X->ir,,X-*H,,X-rn-1X `_IX in -IX yields the short exact sequence of abelian groups cok(i"X) H H"X -» ker(i"_,X) (7) which also represents an element

{H"X} E Ext(ker(i"X),cok(i"X)). (8) Theorem 2.5.10 below shows that the elements in (6) and (8) coincide.

(2.5.9) Addendum Let X be a CW-complex with 1r, X = 0. Using the CW-structure we obtain the following alternative definition of the Postnikov invariant knX; compare VI.8.13 in Baues [AH]. We can choose P"_,X with X" = (P"_ ,X)' in such a way that the attaching map of (n + 1)-cells in P"_ 1X, given by a homomorphism

fn+I : Cn+ I(Pn- IX)-'TnX" as in (2.1.19), is surjective. Then the composition i * fn+ Cn+1(Pn-IX) -. rr,X" -' 1T. X, is a cocycle which represents the Postnikov invariant knX= {r*fn+I} EHn+I(P"-1X,ir X). Here i * is induced by the inclusion X" CX.

(2.5.10) Theorem on Postnikov invariantsWith each 1-connected CW- complex X there is canonically associated a sequence of elements (k3, k4,...) with kn =knXE 143"_,(X,rrnX) such that the following properties are satisfied:

(a) Naturality: for a map F: X - Y we have (1rnF)*(knX) =F*(k"Y) E4n_,(X,1rnY). 2 INVARIANTS OF HOMOTOPY TYPES 59

(b) Compatibility with in X: (knX)* =inXEHom(rnX,irnX).

(d) Vanishing condition: all Postnikov invariants are trivial, that is kn = 0 for n3, if and only if X has the homotopy type of a product of Eilenberg -Mac Lane spaces K(?rn, n), n >_ 2. This theorem is the precise analogue of the corresponding theorem for boundary invariants; see Theorem 2.6.9 below.

Remark Postnikov invariants kn (also called k-invariants) were invented by Postnikov in 1951. From a different point of view J.H.C. Whitehead [GD] studied them in the context of the certain exact sequence. Nowadays Post- nikov invariants are discussed in many textbooks on algebraic topology and homotopy theory. While the naturality of the invariants is a classical result which appears in many textbooks I could not find the compatibility properties of Theorem 2.5.10(b), (c) in the literature. In fact, G.W. Whitehead [EH] in 1978 discussed Postnikov invariants and also the elements u,, and ut associated with a cohomology class u, see (2.1.31); the compatibility properties directly related to these elements are not explained although the certain exact sequence is described. Compare also the remark following Theorem 4.4.4. Proof of Theorem 2.5.10 The naturality is an easy consequence of the description of kn X as a transgression element; see Definition 2.5.8(3). The naturality is also proved in IX.2.6 of G.W. Whitehead [EH]. The vanishing condition (d) is well known; it is an easy consequence of the fibre sequence of Definition 2.5.8(2). We now prove the compatibility with in X. For this we use the explicit construction of k,,X by the attaching map f°+, in Addendum 2.5.9. Recall that Z,, X denotes the cycles in Cn X. We have the commutative diagram

f,tI- Xn Cn+1Pn-1X! 'Tn U rn X ?rnX (2.5.11) =To

Hn+IPn-IX

Here qis the quotient map and µ(kn X) is defined by the cocycle i* f n01 1 60 2 INVARIANTS OF HOMOTOPY TYPES

(representing k,,X) as in (2.1.31)(1). This shows that the outside of the diagram commutes. On the other hand, the Hurewicz homorphism yields the composite h': ir"X" -. H"X" C C"X = CP"_ IX for which h' f,°+ 1is the boundary in C * P"_ ,X. This shows that the inside of the diagram commutes. Hence we get the compatibility (b) which is equivalent to the equation

(2.5.12) (i"X)® = µ(k"X). For the intricate proof of the compatibility (c) we use the properties of the cellular boundary 13 described in (11.4.6) of Baues [CH]. For this we recall the following notation.

(2.5.13) DefinitionFor a chain complex C we consider the commutative diagram

Cn+1-* C. - Cn-1 - I q 1qn-I ... -* 0-'Cn/dCn+]- ''Cn-1 -

Here qn is the quotient map and qi is the identity for i < n. The bottom row is a chain complex C(") which we call the n-type of C. Clearly q: C -> C(") is a chain map which induces isomorphisms q,: H;C - Hi C(") for i:5 n. More- over q is natural with respect to chain maps and chain homotopies.

The cellular map p" -1: X -> P"_ 1X induces the chain map

f3:C=C*XC.P_1,q (C*Pn-1X)(n+1) =P where P is the (n + 1)-type of C * P"_ 1X. Since pn_ Irestricted to n-skeleta is the identity of X" we see that also )9j: C; = Pi is the identity for i:< n. Thus only f3"+ 1:Cn+ I - Pn + 1is relevant. The chain map S represents the cellular boundary invariant explained in more detail in (11.4.6) of Baues [CHI. In particular we prove in (II.4.12)(4) of Baues [CH].

(2.5.14) Lemma For i * f °+ 1in (2.5.11) there is a commutative diagram

Cn+1Pn-IX i-a TrnX

lq" . i a Tg. q Pn+ I cok fan. I

where qn + 1and q are the quotient maps and where O' is an isomorphism. 2 INVARIANTS OF HOMOTOPY TYPES 61

Thus by Addendum 2.5.9 the composite O'q is a cocycle which represents k X; see also (11.4.14) in Baues [CHI. We now consider for the chain map 0 above the cofibre sequence of R in the cofibration category of chain complexes; see Baues [AH]. For this we choose first a factorization /3:CNZs->P of A where C N Zs is a cofibration so that Za and the quotient D = Z,3/C are free. The short exact sequence C N Zs - D induces the long exact homology sequence in the top row of the following commutative diagram

(2.5.15) H., IC0. HHP -0

H+ 1X --->F. X --a7T, ,X --- H X-T;; -1X

The bottom row is given by Whitehead's exact sequence and O' is induced by O' in Lemma 2.5.14. The isomorphisms O are described in Definition 2.5.7(4). Compare 11.4.11 in Baues [CH] where the diagram is also obtained for the case that X is not simply connected. We are going to apply Theorem 2.1.33 for the top row of (2.5.15); this yields the proof for the compatibility (c) of Theorem 2.5.10. For the quotient map r: Z. -, D we have the mapping cone C, as defined in (2.1.32). Moreover by (II.8.24X *) in Baues [AH] the diagram

D , sZs I- `1--

commutes. Here the top row is defined as in (2.1.32)(1) and the bottom row is part of the cofibre sequence for G. Thus we obtain the commutative diagram

Hn+1ZP Hn+1Cr -a' (2.5.16) _1I= a # 1 D H P

Here the top row corresponds to (2.1.32)(2) and the bottom row is given by the top row of (2.5.15). 62 2 INVARIANTS OF HOMOTOPY TYPES

(2.5.17) Lemma There exists a unitary class u in H"+ 1(D, H"+ 1D) such that the composite

H"+1(D H"+1D) r. H"+1(Zjs,Hn+D)=H"+I(p"-1X,H"+1D) satisfies O'*(r*u) = kX E H"+'(P"_ 1X, -rrX). For the proof of the lemma we observe that we can choose Z. such that (Z's)Y"+11 =P(n+1) and D,("+') = cok/3n+1, so that u is represented by D" +-* D,', +i') = cok /3n + 1 - Hn + 1 D. Finally we are ready to apply Theorem 2.1.33 which shows

(2.5.18) - (r*u)t = ( Hn+1Cr) E Ext(H"Z.,cok Hn+1r).

Hence we get, by Lemma 2.5.17, Definition 2.5.13, and (2.5.15) the equivalent equation

(2.5.19) (k,, X)t = {H"X} E Ext(F, _1X,coki"X).

Hence the proof of Theorem 2.5.10 is complete.

We derive from Theorem 2.5.10 on Postnikov invariants the following criterion for products of Eilenberg-Mac Lane spaces.

(2.5.20) Proposition A simply connected space X is homotopy equivalent to a product of Eilenberg-Mac Lane spaces if and only if the Hurewicz homomorphism

h: 7r,, X->H,, X is split injective for all n.

The proposition has a nice dual; see Proposition 2.6.15 below.

Proof Since h is injective we see that i, X: F" X -* ar,, X is trivial, i,, X = 0, for all n. Hence by Theorem 2.5.10(b), (c) we see that k"X = 0{H" X} with { H" X) E Ext(F,,_ 1X, an X) given by the extension

7rnXh N H,, X- rnX.

Since h is split injective we get (H,, X) = 0 and hence knX= 0 for all n. Hence the Proposition is a consequence of Theorem 2.5.10(d)

Remark There is a simple direct proof of Proposition 2.5.20. Using a 2 INVARIANTS OF HOMOTOPY TYPES 63 retraction r": H" X - rr" X of the Hurewicz homorphism we can choose an element

r;, EH"(X,ir"X) _ [X,K(ir,, X, n)]with A(rr) = r" .

The collection of the elements r,,yields a map from X into a product of Eilenberg-Mac Lane spaces which, by the Whitehead theorem, can be seen to be a homotopy equivalence.

(2.5.21) Example There exists a space Y for which the Hurewicz homomorphism h"Y is injective for all n but not split injective. In fact, such a space is the (n + 2)-type Y= PP+2X(217) where s >_ 2 is a power of 2. HereX(2,) is a space in the list in Definition 12.5.3 with homotopy groups ir"Y = 7L, 7r" + 1Y = 0 = H"+1Y,and a" + ZY = 7L/s. The Hurewicz homomorphism satisfies h"+ 2Y:7L/s N 7L/2s. This example shows that the condition `split injective' in Proposition 2.5.20 cannot be replaced by the weaker condition `injective'.

2.6 Boundary invariants of a homotopy type

In this section we introduce fundamental new homotopy invariants of a simply connected CW-space which we call boundary invariants. There are two possible ways of defining these invariants: on the one hand, we obtain them by use of the boundary operator in the f-sequence with coefficients; on the other hand, they can be defined via the Postnikov tower and pseudo- homology groups. We first consider the boundary operator in the f-sequence with coefficients. Let A be an abelian group and let X be a simply connected CW-space. Using the boundary operator bA in (2.3.2) one gets the following commutative diagram with short exact rows.

Ext(A,Hn+1X)N H"(A,X) -µ- Hom(A,H"X)

1(b_ I X). lbw Ext(A, I'"X) f,,_,(A,X)-. Hom(A,f"_,X) (2.6.1) push

Ext(A,cokb"+,X) '-' rm(bA' Hom(A,f _1X) "+1X) 64 2 INVARIANTS OF HOMOTOPY TYPES

The left-hand column is exact. Moreover im(bn+1X)* is the image of the homomorphism (bn+ 1X)*in the diagram and i #is the quotient map; equivalently i # is the projection for the cokernel of A(bn + 1 X) * . On the other handis I'X-' cok isthe projection for the cokernel of bn+ 1X: H,, X -* I' X. Hence i # and i * in the diagram are surjective and one readily checks that the subdiagram `push' is a push-out diagram of abelian groups. Using Definition 2.2.9 one gets the binatural inclusions

r"-1(A,X) rn-1 (A, X) rii-1(A,X) (2.6.2) C C Aim(bn+1X)* iim(bn+1X)* Aim(bn+1X)*

Here and IF,,_ I areactually functorial in A E Ab while rn -1 is not. Let spaces2 be the homotopy category of simply connected CW-spaces. Then the left-hand group in (2.6.2), which we denote by (A, ( 2.6.3) *n I(A,X)=Aim(bn+IX) yields a bifunctor

O,,_ 1: Ab°P x spaces2 -> Ab.

Moreover we have a binatural short exact sequence

Ext(A,cokbi+1X) ln_1(A,X) _W Hom(A,f, _1X) where F,"- 1 X = image(bn X : H X -* Fn_ 1X).

(2.6.4) Definition We define the boundary invariant 6n X, n Z 3, of a simply connected CW-space X as follows. Consider diagram (2.6.1) where we set A = H X. Then we get

rn inXEsfln Mrn(brt+1X)* by

RnX = i#bAµ-1(1H,X).

Here 1H,x EHom(A, H X) is the identity of H X. Since image(b+ 1X)._ kernel(i) we see that the element pnX = i#bA(a) does not depend on the choice of a r=- H,,(A, X) with µ(a) = lynx. The commutativity of the diagram implies the equation (A"X)* =A(PnX)=bnX. 2 INVARIANTS OF HOMOTOPY TYPES 65

This shows that is actually an element of the subgroup 1(H X, X); compare the definition of F."_ ,in Definition 2.2.9. We have chosen the name `boundary invariant' for since, on the one hand, the equation µ( /3 X) = b,, X shows that 8 X is a refinement of the secondary boundary operator b,,X of J.H.C. Whitehead. On the other hand the boundary operator b' in the r-sequence with coefficients is the crucial ingredient in the definition of /3 X.

(2.6.5) Addendum We can define the boundary invariant 0 X by use of the element (/3) E F,,_ X) withz( /3) = b,,Xin Definition 2.3.5(16), namely

13,, X = i 0{ /3 } where i,,is the quotient map in (2.6.1). This follows readily from the definition of the boundary operator b A in terms of (3) in Definition 2.3.5(20). The formula i,,((3) shows the direct connection of 6,, X with the attaching maps in the space X; this connection will be heavily used in proofs below.

Given an element 13 E 1(A, X) we obtain elements 0* =A(0)EHom(A,F._1X) (2.6.6) Qt = 0-1j*( f3) E Ext(ker( /3*),cok(bi+1X)).

Here 0 and µ are given by the binatural short exact sequence in (2.6.3). Moreoverj: ker(/3 *) C Aistheinclusionandj*: e,,_ 1(A, X) - 1(ker(Q* ), X) is induced by the bifunctor 1. As in(2.1.31)(4) one readily checks that the elements in (2.6.6) are well defined. As an example, we obtain, for A = 0,, X, the element

where b X: H,,X --. l,,_ 1X isthesurjective homomorphism given by b,, X: H,, X -+ F,,_ 1X. Moreover we get by (2.6.6) the element

(2.6.7) (8,, X)t E Ext(ker(b,,

which can be compared with the element

(2.6.8) Or,, X) E Ext(ker(b,,

Here is given by the extension

N ir,, X-*ker(b,, X) 66 2 INVARIANTS OF HOMOTOPY TYPES obtained by the exact sequence b_IX Hn+tX H.X b"X X.

The next result shows that the elements (/3 X )t and {if,, X) actually coincide. In fact the next result is the precise analogue of Theorem 2.5.10 on Postnikov invariants. The similarity of Theorem 2.5.10 and the following Theorem 2.6.9 emphasizes the duality between k-invariants and boundary invariants.

(2.6.9) Theorem on boundary invariantsWith each 1-connected CW-complex X there is canonically associated a sequence of elements ('63,/34,...) with

J3 such that the following properties are satisfied:

(a) Naturality: for a map F: X -* Y we have

F* ( 6,X) = E Cn- 1(H,X,Y)

(b) Compatibility with bn X: ()3,X)*

(d) Vanishing condition: all boundary invariants are trivial, that isNn =0 for n >_ 3, if and only if X has the homotopy type of a one point union of Moore spaces M(H,, n), n >_ 2, with H,, = H,(X).

Proof The naturality is an easy consequence of the naturality of the operators i # and b and µ. In fact

F* F,K i#b,, A-1(1H"X)

=i#bnµ-1((Hn F)*1H"X

=i#b,, µ-1((H,F)*1H"Y)

= (H,F)*i#bnA-1(1H"Y) = (H,F)*J3,Y. 2 INVARIANTS OF HOMOTOPY TYPES 67

Compatibility with b, ,X was already obtained in Definition 2.6.4. Finally we prove compatibility with {ir" X} by the definition of 0" X in Addendum 2.6.5 above. We use the notation in Definition 2.3.5. We have the commutative diagram d P B" Z. -'H" Ui U; 11 B" N kerb" p) - kerb"

with exact rows. The inclusion i yields a map i for which the following diagram homotopy commutes M(B", n) ?q M(K,n-1) M(H",n-1) 9Xrt-1 U U U X"-z Here q is the quotient map for M(K, n - 1) = C. Since by definition of b in Definition 2.3.5(6) we know ibj = 0 there is a map it for which the composition

7r> B" 7r"X"-` - F, -> cok b"+1 represents {7r}=0-11*(i3+1X) EL 11*(Yn+1X). (1) We show that also represents the extension class {ir"}. Let X be the mapping cone of the restriction u I M(z,,,, n) v M(tB _,, n -1),see Definition 2.3.5(9) and (7). Then 8 in Definition 2.3.5(15) yields the map 13:M(Hn,n-1)-'Xn-1 cX with Cs=X"'. Therefore we have the following diagram of cofibre sequences M(K,n-1) M(B",n)-0 M(L,n)-4 M(K,n)

l1 l- I lyt M(H,,,n-1)s X -b Cs 4M(H",n) 68 2 INVARIANTS OF HOMOTOPY TYPES

We apply the functor irn to this diagram and we get the commutative diagram

B" N L-» K

(2) 1rnX -b irn -h H.

Here it"C,, = 7r,, X"+' = ar" is the homotopy group of X and h = it"(q) is the Hurewicz map. Therefore we have the group

h (irn) = ker bn = K. Since the top row of (2) is a free presentation of this group the map jir: Bn -* ker h represents {7r"), see Lemma 2.1.30. In fact, fir is the same as it in (1). This completes the proof of (c). Finally the definition of the boundary invariants shows that all j3n(X) are trivial if X is a one-point union of Moore spaces; here we use the definition in Addendum 2.6.5 and Definition 2.3.5. On the other hand, the classification theorem in Chapter 3 below yields the inverse; namely the vanishing of allj3, J implies that X has the homotopy type of a one-point union of Moore spaces. This completes the proof of Theorem 2.6.9.

In our second definition of the boundary invariants we use the (n - 1)-type

(2.6.10) PX 1:X-Pn-1X of the simply connected space X; see Definition 2.5.6. Using pseudo- homology groups with coefficients in A we obtain the following diagram which resembles diagram (2.6.1); in fact we show below that the following diagram is naturally embedded in diagram (2.6.1).

(2.6.11)

Ext(A, Hn+1X) H,,(A,X) Hom(A,H"X)

X IM", I y(P; i).

-µ+Hom(A,H"P"_,X)111I Ext(A, Hn+1Pn-1X) H,,(A,P,, -1X)

push II

x H"(A,P"X) Ext(A,cok Hn+1Pn-1)N Hom(A,H"Pn_jX) Aim(Hn+ 1Pn-1). 2 INVARIANTS OF HOMOTOPY TYPES 69

The rows are short exact and the left-hand column is exact. Moreover im(HR+ 1ph I)*is the image of the homomorphism (Hn + i Pn I) *in the diagram and i# is the quotient map; equivalently i, is the projection for the cokernelof thecompositionz( Hn+ 1p,-d,,. On theotherhand, i s HR+PR_ 1X -* cok Hn isthe for 1 + 1p, , projection thecokernelof Hn + 1 Px I: HR + 1 X -HR + 1 PR- IX. Hence i * and iin the diagram are surjective and hence the subdiagram `push' in (2.6.11) is a push-out diagram of abelian groups. The naturality of the (n - 1)-type (2.6.10) shows that the group HR(A,P.-1X)

(2612) c' 1(A ,X) = n Aim(HR+1Pn-1)x yields a functor R -1: Ab°P x spaces, - Ab together with a binatural short exact sequence given by the bottom row of (2.6.11).

(2.6.13) Definition We define the (homological) boundary invariant f3nX, n >_ 3, of a simply connected CW-space X as follows. Consider diagram (2.6.11) where we set A = Hn X. Then we get X) HR (A,PR 1 f3rtXii-1(A, X) = X A im(Hn+IPn-1) f3rtX=i#(Pn 1)*,u-1(lx,x).

Here 1H,x E Hom(A, HR X) is the identity of HX. As in Definition 2.6.4 we see that f3;, X is a well-defined element. The naturality of the diagram (2.6.11) immediately yields the naturality of the invariant A,, X, that is, for f : X -* Y in spaces2 we have f( fart X) = f *( f3rtY). Hence on 'X is a well-defined invariant of the homotopy type of X. The next result shows that we can identify the homological boundary invariant 9,,X with the boundary invariant $, X in Definition 2.6.4.

(2.6.14) TheoremFor a simply connected CW-space X and an abelian group A there is a binatural isomorphism 0: Z'-,(A, X)=Cn_1(A, X) compatible with 0 and A. Moreover for A = HR X the isomorphism O carries the homological boundary invariant 6,,X to the boundary invariant /3n X, that is 0(13;,X)= f3RX. 70 2 INVARIANTS OF HOMOTOPY TYPES

Proof By Definition 2.5.7 there are natural isomorphisms (1)

(2) such that the diagram

HH+1X

(3) H +IX 1' X commutes. Hence (b + 1 X) *in (2.6.1) corresponds to (H+ 1Px 1)*in (2.6.11). We now show that in addition there is a binatural isomorphism (4) such that the following diagram with short exact rows commutes

Ext(A,Hi+1PiX) H (5) 110. Ile 11 0. Ext(A,FX) > F,_1(A,X) Hom(A,F,_1X)

Moreover O in (4) makes the diagram

1b^ 11 9 (6) 1;;-,(A, X) commutative, where b'' is the boundary operator in (2.6.1). Hence (3), (5), and (6) show that 0 in (4) induces an isomorphism as in the Theorem for which 0,6'X = 13 X, this is readily seen by comparing diagram (2.6.1) with diagram (2.6.11). We obtain 0 in (4) as follows. Using the boundary operator bA in (2.3.2) and Definition 2.3.5 for the space Y = P - , X we obtain the commutative diagram with short exact rows: Ext(A, H (A,Y)-* Hom(A,

J1I0_,Y). IbA pull 10,Y). Hom(A,I'i_2Y) = =1a. =1(r.-,P). Hom(A,1'i_1X) 2 INVARIANTS OF HOMOTOPY TYPES 71

Here p =p' ,: X --> Y is the (n - 1)-type of X in (2.6.10). We know that p induces isomorphisms r p, p,,. and r_ , p asshown in the diagram. Also b A in the diagram is injective by (2.3.2) since 1r"(A,Y) = 0. Moreover is an isomorphism and is injective by Definition 2.5.7(3). This shows that the top part of the diagram is a pull-back diagram. Hence we obtain the isomorphism © in (4) by the composite (p *)-' bA In fact, by (1) the injection above corresponds to the inclusion F,1 X c r_ ,X since the diagram

H"Y > --> r"_,Y

ell IIr-P F,"-,X C r,,-,x commutes; compare the definition of r,"_ , in Definition 2.2.9. The definition of O by (p *)-' bA shows that diagram (5) and diagram (6) commute.

Finally we derive from the theorem on boundary invariants the following criterion for one-point unions of Moore spaces.

(2.6.15) Proposition A simply connected space X is homotopy equivalent to a one point union of Moore spaces if and only if the Hurewicz homomorphism

h: is split surjective for all n.

This is the precise dual of the criterion for products of Eilenberg-Mac Lane spaces in Proposition 2.5.20.

Proof Since h is surjective we see that b X: H X - r_ ,X is trivial, b X = 0, for all n. Hence by Theorem 2.6.9(b), (c) we get with E r,, X) given by the extension

h I,,x>1rX HHX. Since h is split surjective we get {1r X} = 0 and hence 0,, X = 0 for all n. Hence the proposition follows from Theorem 2.6.9(d). Remark There is a simple direct proof of Proposition 2.6.15. Using a splitting s": H. X --> 1r X of the Hurewicz homomorphism we can choose an element

(s;, E withlL(s;,) =s,,. 72 2 INVARIANTS OF HOMOTOPY TYPES

The collection of these elements s' yields a map from a one-point union of Moore spaces to X which, by the Whitehead theorem, can be seen to be a homotopy equivalence. (2.6.16) Example There exists a space X for which the Hurewicz homomorphism h,,X is surjective for all n but not split surjective. In fact, such a space is given by X = "- ' RP , n > 4, as we show in (8.1.11). Hence the condition `split' in Proposition 2.6.15 is necessary.

2.7 Homotopy decomposition and homology decomposition In this section we describe both the `homotopy decomposition' and the `homology decomposition' of a simply connected CW-space. These concepts are Eckmann-Hilton dual to each other. The homotopy decomposition is also called the Postnikov decomposition or Postnikov tower associated with a space X. The homology decomposition was obtained by Eckmann and Hilton, and Brown and Copeland. The `homotopy decomposition' describes a construction of spaces using Eilenberg-Mac Lane spaces K(ir, n) as building blocks. This is explained more precisely in the following definition and theorem. (2.7.1) Definition A homotopy decomposition (1-connected) is a system of the form

Here 7x2,113, ... is a sequence of abelian groups and X2, X3,... is a sequence of CW-spaces which fit into the following diagram (n >_ 3).

K(7rn, n) Xn K(7rnI, n + 2) I K(trn_1,n - 1)- 1)

, K(7r3, 3) -> X3 K(7r4, 5)

1 K(7rz, 2) = Xz -- K(7r3, 4) Each vertical arrow is a fibration in Top such that, for n z 3, the sequences K(7rn,n)=+Xn (2) 2 INVARIANTS OF HOMOTOPY TYPES 73 are fibre sequences. That is, q is a principal fibration with classifying map k and fibre K(7r,,, n) = fZK(7r,,, n + 1). The classifying map k represents the cohomology class k,, E=- H" +(X.- 1,7r.) (3) which, for n >_ 3, yields the sequence of elements k3, k4,... in(1). The fibre sequences (2) imply that 7rjX = 0 for j > n so that X,, is a simply connected n-type. Hence the map k,,: X,,_ 1-* K(7r,,, n + 1) is 7r * -trivial, that is, the map k induces the trivial homomorphism on homotopy groups. Let be the inverse limit in Top of the sequence of fibrations X2 .- X3

(2.7.2) TheoremFor each simply connected CW-space X there exists a homotopy decomposition, together with a map f : X - lim (X,,), which induces isomorphisms of homotopy groups f.: 7r X = 7r,,. 4---

Compare, for example, G.W. Whitehead [EH]. The theorem shows that the homotopy type of X is determined by the homotopy decomposition of X which essentially is unique. Moreover an important feature of the homotopy decomposition isitsnaturality with respect to maps X-*X'; see G.W. Whitehead [EH] or Baues [OT]. In particular the homotopy type of X in Theorem 2.7.2 is an invariant of the homotopy type of X.

Proof of Theorem 2.7.2 We replace inductively the maps P X --> P, _ 1 X in Definition 2.5.6 by fibrations so that we get a sequence of fibrations as in Definition 2.7.1 together with commutative diagrams

P IX P,, X =1 * 1 xn xn-1 in which the vertical arrows are homotopy equivalences. Moreover we can choose inductively maps f,,: X -> X with q f = f such that X -p P X - X given by p,, in Definition 2.5.6 is homotopic to f,,. Then induces the weak equivalence f in the statement of the theorem. The naturality of the homotopy decomposition can be derived from the functorial properties of the Postnikov section X - P X in Definition 2.5.6 and the naturality of the Postnikov invariants in Theorem 2.5.10. Next we explain the concept of a homology decomposition which uses Moore spaces M(H,,, n) as building blocks.

(2.73) Definition A homology decomposition (1-connected) is a system of the form

(1) 74 2 INVARIANTS OF HOMOTOPY TYPES

Here Hz, H3,...is a sequence of abelian groups and X2, X3,...is a sequence of CW-spaces which fit into the following diagram (n z 3).

"" M(H,,, M(H,,,n)"- X ,n) I

IM(H,,,n-1) I

M(H3,3)E- X3 --M(H4,3) I M(H2, 2) = X2 -k' M(H3, 2)

Each vertical arrow is a cofibration in Top such that, for n >- 3, the sequences

M(H,,,n)+-X"4 (2) are cofibre sequences. That is, in is a principal cofibration with attaching map k;, and cofibre M(H, n) = I M(H,,, n - 1); equivalently X is the mapping cone of the map k;,. This map represents the homotopy class

k;, E (3) which, for n >- 3, yields the sequence of elements k3, k4',... in (1). We require that these elements are H,, -trivial, that is, the map k', induces the trivial homomorphism on homology groups. Let be the direct limit in Top of the sequence of cofibrations X2 >-+ X3 >-->

(2.7.4) TheoremFor each simply connected CW-space X there exists a homology decomposition, together with a map f : lim X, which induces isomorphisms of homology groups f.: H,, _- H,, X.

Since the direct limit in Theorem 2.7.4 is a CW-space the map f is actually a homotopy equivalence. Thus we may construct any 1-connected homotopy type with homology groups H2, H3, ... by a process of successively attaching cones CM(H,,, n - 1) via homologically trivial maps. Conversely, any such construction produces a homotopy type with homology groups H2, H3,... . The homotopy classes k' of the attaching maps in Definition 2.7.3(3) are `k'-invariants' of the homology decomposition. These are dual to the k- invariants in the Postnikov decomposition. In fact, the homology decomposition was introduced by Eckmann and Hilton as a dual of the Postnikov 2 INVARIANTS OF HOMOTOPY TYPES 75 decomposition. The homology decomposition turned out to have a disadvantage, namely it fails to be natural. In particular, the homotopy type of the section Xn of a homology decomposition is not an invariant of the homotopy type of X. Therefore, also, the k' invariants do not have the desired property of naturality. For this reason we have introduced in Section 2.6 above new invariants which we call the boundary invariants of X. The boundary invariants can be thought of as being exactly the ingredient of the k'-invariants which is natural. Moreover, these boundary invariants determine the homotopy type in the same way as the Postnikov invariants. We shall prove this in Chapter 3. For the proof of Theorem 2.7.4 we use the principal reduction in Definition 2.3.5(4). This proof also shows how the attaching map k,, above is related to the boundary invariant 13 X.

Proof of Theorem 2.7.4Let X be a 1-connected CW-complex. We may assume that X' = *. We choose splittings t for the short exact sequences

d Zn_Cn-'Bn-1 where Bn_ 1 =dC, C = C* X. Then we have C,, = tBn ® Z,,. The attaching map f of n-cells in X yields maps fB and f2 for which the following diagram commutes

M(Zn, n - 1) C M(C,, n - 1) J M(tB"_ 1, n -1) f If Xn- 1

Let X,_be the mapping cone of fB and letis X"-' cXn_ 1be the 1 inclusion. Assume now Xn_ 1admits a homology decomposition as in the theorem. Then X,, admits one also since Xn is homotopy equivalent to the mapping cone of a map k' =i(3: M(H,,, n - 1) )Xn_1 which is homologically trivial. We obtain /3 as in Definition 2.3.5(15) by the map v: V - X"-' with mapping cone C,. = X"+' This completes the proof of Theorem 2.7.4.

Remark The map /3 in the proof above is also used for the construction of the boundary invariant 6,,X in addendum (2.6.8). On the other hand, we obtain6,,X directly by the homology decomposition as follows. Let X = lim X. be given by a homology decomposition as in Definition 2.7.3. Then X can be chosen to be a CW-complex for which the skeleta X" satisfy

Xn-1 CXn_1 CX". 76 2 INVARIANTS OF HOMOTOPY TYPES

Moreover k;, admits a factorization R for which the following diagram homotopy commutes.

X"-, U

(2.7.5) M(H",n - 1) =a X' 1 U U

M(Z",n-1) -X"-2

Hence 0 represents an element 13"X in Z"_ 1(H", X) by definition of I'"_,(A,X) in Definition 2.2.3; see Definition 2.2.9, Lemma 2.3.4 and the definition of C"_ ,(A, X) in (2.6.2). Hence diagram (2.7.5) describes exactly how to deduce from k', the boundary invariant 6,,X. We would like to emphasize that the classical concepts of homotopy decomposition and homology decomposition above do not have a strong impact on the classification of homotopy types. They only provide instructions on how to build a 1-connected CW-space with prescribed homotopy groups and homology groups respectively. The main problem is to decide whether two such constructions are actually homotopy equivalent. A further deep problem is connected with the relationship between the homotopy decomposition and the homology decomposition of a given space X. In fact, the homotopy decomposition of a Moore space M(A, n) (for example a sphere) involves the computation of all homotopy groups of the Moore space. Conversely the homology decomposition of an Eilenberg-Mac Lane space K(A, n) requires the computation of all homology groups of K(A, n) as achieved by the work of Eilenberg and Mac Lane, and Cartan. A basic result concerning the connection between the homotopy decomposition and the homology decomposition is given by the compatibility properties of Postnikov invariants and boundary invariants respectively; see Theorem 2.6.9(b, c) and Theorem 2.5.10(b, c). Rationally, that is for 1-connected rational spaces X, the connection between the homotopy decomposition and the homology decomposition of X is completely understood; see Baues and Lemaire [MM].

2.8 Unitary invariants of a homotopy type

We here describe the `unitary invariants' of a homotopy type which were introduced in (V.1.7) of G.W. Whitehead [RA]. The definition of these invariants is similar to the definition of the boundary invariantsjB,, in Definition 2.6.13. We show, however, that unitary invariants are not suitable for the classification of homotopy types; in fact there are spaces which can be distinguished by boundary invariants but not by unitary invariants. 2 INVARIANTS OF HOMOTOPY TYPES 77

In the definition of unitary invariants we use again the (n - 1)-type

(2.8.1) of the simply connected space X. Using the cohomology groups with coefficients in an abelian group A we obtain the following diagram which resembles (2.6.11).

(2.8.2)

A) - Hn+t(PnX, A)-. Hom(Hn+,P,,1X, A)

A) > Hn+'(X, A)- Hom(H,+ 1X, A)

push

Hn+1(X, A)* Ext(kerb,,X, A) >-* Hom(Hn+1X, A) Dim(HnPn_I)

The rows are given by the short exact coefficient sequences. For the inclusion j: ker(bn X) c Hn X the left-hand column is exact since we can identify

Hn X H°P°' Hn Pn_ 1X

9 (1)

11 6-X,F, 1X Cr, 1X

Moreover we can identify

Hn+ 1XH Hn+1Pn_IX

II6 (2) II b,X Hn+1X rfX

Compare the proof of Theorem 2.6.14. Let

im(HnPn ,)*=(H p, 1)*Ext(H.P._1X, A) be the image of the homomorphism (Hn px )* in diagram (2.8.2) and let j* be the quotient map; equivalently j* is the projection for the cokernel of the 78 2 INVARIANTS OF HOMOTOPY TYPES composite 0(H" px , ). Hence j* and j# are surjective and one readily checks that the subdiagram `push' in (2.8.2) is a push-out diagram of abelian groups. The naturality of the (n - 1)-type (2.8.1) shows that the group Hn+'(X, A) (283) 21n+ '(X A)= yields a functor 21"+ I : spacesOp x Ab Ab, together with a binatural short exact sequence given by the bottom row of (2.8.2).

(2.8.4) Definition We define the unitary invariants u'+ '(X), n >_ 2, of a simply connected CW-space X as follows. Consider diagram (2.8.2) where we set A = H,,+, P"_ ,X1'" X. Then we get H"+'(X,1'"X) u"+I(X) E 21n+I(X,f X) _

un+I(X) =j*(Pn I)*9-I(1r^x).

Here 1rrtx EHom(Fn X, A) = Hom(Hn+ IPn_ ,X, A) is the identity of I'n X, hence an element u E µ-'(1rTx) is a unitary class (see (1.2.32)), which via j#(PxI)* yields the invariant u'+ 1M. One readily checks that u"+ 1(X) is well defined. Given an element u E ll"+'(X, A) we obtain elements u * = µ(u) E Hom(H"+ I X, A) (2.8.5) u t = 0 -' q * (u) E Ext(ker bn X, cok(u * )) .

Here A and µ are defined by the binatural short exact sequence in (2.8.2). Moreover q: A - cok(u *) is the quotient map and

q*:21"+'(X, A) - 21n+'(X,cok(u* )) is induced by the bifunctor 21"+' As in (2.1.31X4) one readily checks that the elements in (2.8.5) are well defined. As an example we obtain for u = u` 1X the element (U"+IX)* =A(un+IX)=bn+IX and the element

(u"+' X )t E Ext(ker bn X, cok bn+ ,X) which can be compared with (1rnX) in (2.6.8). 2 INVARIANTS OF HOMOTOPY TYPES 79

(2.8.6) Theorem on unitary invariantsWith each 1-connected CW-complex X there is canonically associated a sequence of elements (u4, u5, ...) with

un+ I = un+ IX E 21 n+' (X, r" X),n > 3, such that the following properties are satisfied:

(a) Naturality: for a map F: X --> Y we have

F*(un+ 1Y)= (rnF) * (un+1X) E can+I(X, I'nY).

(b) Compatibility with b,,- IX: (un+1X)* =bn+1XEHom(Hn+1X,rnX)

-(u"+'X)t = {irnX} E Ext(ker(b"X),cok(b"+1X)).

(d) Vanishing condition: all unitary invariants are trivial, that is u"+' = 0 for n - 2, if and only if X has the homotopy type of a one point union of Moore spaces M(H, n), n > 2, H,, = H"(X).

Proof Propositions (a) and (b) are clear by construction. Moreover (c) is a consequence of Theorem 2.1.33, compare (V.1.7) in G.W. Whitehead [RA]. Now assume u"+'X = 0 for n > 3. Then bn+ I X = 0 for all n and also {i7, X) = 0. Hence h: 7rn X -' H,, X is split surjective and therefore we obtain (d) by Proposition 2.6.15. 0

The theorem shows that unitary invariants have almost the same properties as boundary invariants in Theorem 2.6.9. The unitary invariants, however, are not suitable for classification as we show by the following example.

(2.8.7) ExampleLet X and Y be (n - 1)-connected (n + 3)-dimensional spaces defined as follows. The space X is the mapping cone of the map

i17l,,q+i277: M(7L/2,n + 1) -S" VM(7L/r, n).

Here q: M(7L/2, n + 1) --* Sn+2 is the pinch map and q"2: Sn+2 -* S" is the double Hopf map. Moreover i1, i2 are the inclusions and

n = %2: M(7L/2, n + 1) --* M(7L/r, n) is a map which is non-trivial on the (n + 1)-skeleton. Let Y be the mapping 80 2 INVARIANTS OF HOMOTOPY TYPES cone of 71. The homology of X and S" V Y coincide and we have the injective maps

bn+2(X) = bn+2(S" V Y): Hn+2X = 7 1 / 2N I;,_,X = 71/2 ® 71/2 with H"X=71®71/r and fn_,X=H,,(X)®Z/2. Since Hrt+3X=0 the injectivity implies that un+3(X) = un+3(S" V Y) = 0. Hence the unitary invariants do not suffice to distinguish between X and S" V Y. In Chapter 8 we show via boundary invariants that X and S" V Y are not homotopy equivalent. In fact, we have X =X(e2rl,) and Y=X(2rt,) in the notation of Chapter 12, and X is not decomposable.

(2.8.8) RemarkIf X is (n - 1)-connected, n >- 2, then un+2XEHn+2(X,f,H,X) is the Pontrjagin-Steenrod element; compare (V.1.9) in G.W. Whitehead [RA] and (1.6.8) in Baues [CHI. Thus formula (5.3.5) below can be derived from Theorem 2.8.6(c).

(2.8.9) Remark A crucial property of the bifunctors 3n _ ,in Definition

2.5.7 and Cn _ Iin (2.6.3) is the fact that the (n - 1)-type pn 1: X - Pn_ 1X induces isomorphisms

(pn1)*: n-1(Pn-1X,A)=' rt_1(X,A), (p,_ 1)*: These isomorphisms allow the definition of the detecting functor in the classification theorem (3.4.4). For the functor 1"+' we do not have such an isomorphism. 3 ON THE CLASSIFICATION OF HOMOTOPY TYPES

In this chapter we describe fundamental new results on the classification of homotopy types. On the one hand, we get a classification by Postnikov invariants (k-invariants); on the other hand, we obtain a classification by boundary invariants. The general properties of these invariants lead us to introduce algebraic concepts which we call `kype functors' E and `hype functors' F, respectively. Here kype is a new word derived from the words k-invariant and type and similarly hype is derived from boundary invariant and type. A kype functor E and a bype functor F determine categories which we denote by

Kypes(E),kypes(E) and Bypes(F),bypes(F).

Our classification theorem shows that the objects of such categories are models of homotopy types. Hence a classification of homotopy types can be achieved by the computation of kype functors and bype functors, respectively. In later chapters we describe various examples of such computations which lead to optimal algebraic descriptions of certain classes of homotopy types. As a simple example we obtain the old results of J.H.C. Whitehead on the classification of 1-connected 4-dimensional homotopy types, see Section 3.5. The classification theorem of this chapter is the core of the book. We shall describe many applications and explicit examples of this theorem.

3.1 kype functors

The properties of k-invariants of a simply connected CW-space lead to the notion of `kype'. Here kype is an amalgamation of the words k-invariant and type. In Section 3.4 we describe a classification theorem which shows that kypes are fundamental models of homotopy types.

(3.1.1) DefinitionLet C be a category. A kype functor on C is a functor

E:C°PxAb->Ab (1) 82 3 CLASSIFICATION OF HOMOTOPY TYPES together with a binatural short exact sequence (X E C, IT (=- Ab)

Ext(EoX, ,7r) E(X, Tr)-µ*Hom(E1X, ir) (2) where E0, E,: C -> Ab are functors. We call (Eo, El, 0, µ) the `kype structure' of the functor E. We say that that the kype functor E is split if the exact sequence (2) admits a binatural splitting in Ab. In this case E is completely determined by the pair of functors (E0, E1) since we can identify

E(X, Tr) = Ext(E0X, ir) ® Hom(E1X, ir) (3) by the splitting of (2). Moreover we say that E is semitriuial if E0 = 0 is the trivial functor. Clearly in this case E is determined by El. All kype functors E are semisplit in the following sense: for each object X E C there is a splitting homomorphism

sX: Hom(E1X, 7r) - E(X,TT) (4) with usx = 1 which is natural in 7r; that is for cp:it -> ?r' in Ab we have (P * sX = sx cp * . We obtain sX since the functor Ab -> Ab carrying Tr to Hom(E1 X, ir) is projective in the functor category of functors Ab -+ Ab. In Section 3.5 we describe examples of semitrivial kype functors and in Section 3.6 we deal with split kype functors. Using a kype functor E we introduce the following category of E-kypes which is a kind of extended Grothendieck construction, see Remark 3.1.3.

(3.1.2) DefinitionLet E be a kype functor on the category C. An E-kype X=(X,ir,k,H,b) is a tuple consisting of an object X in C, abelian groups it and H, and elements kEE(X,vr), b E Hom(H, E, X) such that the sequence µ(k) H E,(X) IT (1) is exact. A morphism

(f, gyp,4PH): between such E-kypes is given by a morphism f: X -+X' in C and by 3 CLASSIFICATION OF HOMOTOPY TYPES 83 homomorphisms (p: it - 7r', (pH: H -> H' such that the following properties (2) and (3), are satisfied:

f *(k') = p*(k) E E(X, Tr') (2) Here the induced homomorphisms f. E(X',,7r') EM 7r') EM 1T) are given by the bifunctor E. Moreover the diagram

H E,(X)

If* =El(f) (3) H' E,(X') commutes. E-kypes and such morphisms form the `category of E-kypes'. If the kype functor E is clear from the context we call an E-kype simply a kype.We say that an E-kype (X, ar, k, H, b) is injective if b is an injective homomorphism. Let kypes(C, E) (4) be the full subcategoryofinjective E-kypes. Moreover anE-kype (X, ir, k, H, b) is free if H is a free abelian group. Let Kypes(C, E) (5) be the full subcategory of such free E-kypes. We have the forgetful functor 0: Kypes(C, E) -> kypes(C, E) (6) which carries the free E-kype (X, ,;r, k, H, b) to theinjective E-kype (X, Tr, k, H', b') where H' is the image of b and where b' is the inclusion of this image.

Lemma The functor 0 is full and representative.

ProofItis clear that each E-kype has a realization by choosing a surjection H -. ker µ(k) where H is free abelian. Hence 0 is representative. Moreover 0 is full since, by Definition 3.1.2 (2) the homomorphism f. carries ker µ(k) to ker µ(k') and hence we can choose a homomorphism KPH for which Definition 3.1.2 (3) commutes since H is free abelian. 0

(3.13) Remark Any bifunctor E:C°PxK-*L (1) 84 3 CLASSIFICATION OF HOMOTOPY TYPES yields a category Gro(E) which is called the Grothendieck construction on E. Objects are triples (X, ar, k) where X E Ob(C), it E Ob(K), and

kEE(X,Tr). (2) Morphisms (f, cp): (X, ?r, k) --> (X', ir', k') are given by morphisms f: X -* X' in C and (p: Tr -- Tr' in K such that the equation

f *(k') = cp* (k) E E(X, ir') (3) is satisfied; see Definition 3.1.2(2) above. Hence the category of E-kypes above is a kind of enriched Grothendieck construction. Moreover we have, for a kype functor E, the forgetful functor

* : kypes(C, E) - -> Gro(E) which carries the injective E-kype (X, ir, k, H, b) to (X, vr, k). This functor +y is easily seen to be an equivalence of categories. This way we identify an injective E-kype with an object in the Grothendieck construction of E. (3.1.4) Remark Let E be a kype functor on C and let a: K -* C be a functor. Then we obtain a kype functor a *E on K as follows. We define a *E by the composite

a*E: K°P XAb0Xi_C°P xAb_Ab (1) and we obtain the kype structure of a *E by (E0 a, E1 a, 0, µ) where (E0, E 0, µ) is the kype structure of E. Moreover a induces functors

a: kypes(K,a*E)-> kypes(C, E) (2) a: Kypes(K,a*E) Kypes(C, E) (3) which carry (Y, ?r, k, H, b) to (a Y, ir, k, H, K The induced functors a in (2) and (3) are equivalences of categories if a is an equivalence of categories. _We now use the kype structure of the functor E in an essential way. Let X = (X, ir, k, H, b) be an E-kype as above. Then we obtain elements

k* = A(k) E Hom(E1X, ar) (3.1.5) kt = A-'q*(k) E Ext(EOX,cok(k*)). as follows. The homomorphism k * = µ(k) is given by k in X and by the natural transformation µ in Definition 3.1.1. For the quotient map

q: ir-» 7r' =cok(k*) (1) 3 CLASSIFICATION OF HOMOTOPY TYPES 85

(to the cokernel of k *) we get via naturality a commutative diagram

Ext(EoX,ir) N E(X,ar) - Hom(E,X,ir) 1q. Iq. Iq. (2) Ext(E0X, 7T 1) N E(X, ir') -µ-> Hom(E1X, IT')

Here q*p(k) = 0 implies that the element kt = 0-'q*(k) in (3.1.5) is well defined. Let

cok(k*): ` >H(kt) b°»EoX (3) be an extension of abelian groups which represents the element k-,.. Hence we obtain by X the exact sequence

b H___4 E, X ' H(kt)bEo X -> 0 (4) which we call the f-sequence of the E-kype X. This sequence is in the following sense natural with respect to morphisms between E-kypes. We say that a morphism f: X --> X' together with a commutative diagram in Ab H - E,X -* 7r -*H(kt)-4 E0X --0

I If. I if. (5) H'-E,X'-->7r'-*H(kt)- E0X'-+0 isa weak morphism between f-sequences if thereis a homomorphism H(kt) ---> H(kt) which extends the diagram commutatively. Similarly we define a weak isomorphism. Each morphism (f, gyp, !pH) between E-kypes clearly induces a weak morphism between f-sequences for which the vertical arrows in (5) are lpH, E,(f ),cp, and Eo(f ), respectively.

3.2 bype functors

The properties of boundary invariants of a simply connected CW-space give rise to the notion of `bype'; here bype is the amalgamation of the words boundary invariant and type. A hype is the true Eckmann-Hilton dual of a kype discussed in Section 3.1. To stress the duality between bypes and kypes this section is organized in the same manner as Section 3.1. Indeed the strength of this duality is amazing and new; see also Section 3.3. In Section 3.4 we describe a classification theorem which shows that bypes are fundamental new models of homotopy types where bypes are related to homology groups similarly to the way kypes are related to homotopy groups. 86 3 CLASSIFICATION OF HOMOTOPY TYPES

(3.2.1) DefinitionLet C be a category. A hype functor F on C is a functor F:Ab°PxC-->Ab (1) together with a binatural short exact sequence (X E C, H E Ab)

Ext(H, F1 X) N F(H, X) Hom(H, Fo X) (2) where F0, F1: C -> Ab are functors. We call (F0, F,, 0, µ) the `bype structure' of the functor F. We say that the bype functor F is split if the exact sequence (2) admits a binatural splitting in Ab. In this case F is completely determined by the pair of functors (F(,,F1) since we can identify F(H, X) = Ext(H, F, X) e Hom(H, Fo X) (3) by the splitting of (2). Moreover we say that F is semitrivial if F0 = 0 is the trivial functor. Clearly in this case F is determined by F1. Each bype functor F is semisplit in the following sense: for each object X E C there is a splitting homomorphism sx: Hom(H,FoX) -->F(H, X) (4) with ,2sx = 1 which is natural in H E Ab. For this we use a similar argument to that in Definition 3.1.1 (4).

We study bype functors in more detail in Section 3.3 where we show that they are actually `equivalent' to kype functors by a duality isomorphism. We consider an example of a semitrivial bype functor in Section 5 and in Section 6 we describe an example of a split bype functor. Up to a few changes a bype functor is the exact analogue of a kype functor in Definition 3.1.1. We now introduce categories of bypes and Bypes respectively which are the analogues of the corresponding categories of kypes and Kypes in Section 3.1. In Sections 3.5 and 3.6 we describe such categories in case the bype (resp. kype) functors are semitrivial or split. It is interesting to have these simple cases in mind when reading the following definitions.

(3.2.2) DefinitionLet F be a bype functor on the category C. An F-bype X=(X,Ho,H1,b,,O) is a tuple consisting of an object X in C, abelian groups H0, H1, and elements b E Hom(H,, F1X), /3 E F(H0, X, b) with the following properties. Using the homomorphism b and the bype 3 CLASSIFICATION OF HOMOTOPY TYPES 87 structure of F we define the abelian group F(H0, X, b) by the push-out diagram

Ext(H0, Fl X) NF(Ho,X) -µ- Hom(Ho,F0X)

Ii. push II 1

Ext(Ho,F,X/K) » F(HO,X,b)IU Hom(Ho,F0X)

Here is F,X -. cok(b) =F,X/K is the quotient map for the cokernel of b with K = image(b). The element R E F(Ho, X, b) has the property that the sequence µcs) Ho FOX-->0 (1) is exact, that is µ( (3) is surjective. A morphism

(f, wo, API ): (X, Ho, H1, b, !3) , (X', Hl', b', 83') between such F-bypes is given by a morphism f: X - X' in C and by homomorphisms po: Ho - Ho, cp,: HI -+ Hl' such that the following properties (2) and (3), are satisfied. The diagram

I1P, (2)

H; is commutative. Hence f* induces a homomorphism f,,: cok(b) -p cok(b') between cokernels so that the bype functor F yields induced homomorphisms

F(Ho, X, b) f;. F(H0, X', b') P0* F(HO, X', b') with po(p') EF(H0,X',b'). (3)

F-bypes and such morphisms form the `category of F-bypes'. If the bype functor F is clear from the context we call an F-bype also simply a bype. We say that an F-bype (X, H0, H b, 0) is injective if b is an injective homomorphism. Let bypes(C, F) (4) 88 3 CLASSIFICATION OF HOMOTOPY TYPES bethefullsubcategoryofinjectiveF-bypes.Moreover anF-bype (X, H0, H1, b, /3) is free if H, is a free abelian group. Let

Bypes(C, F) (5) be the full subcategory of free F-bypes. We have the forgetful functor

¢: Bypes(C, F) ---> bypes(C, F) (6) which carries the free F-bype (X, H0, H,, b, )3) to the injective F-bype (X, H0, H b', /3) where H,' is the image of b and where b' is the inclusion of this image. Hence cokernels cok(b) = cok(b') coincide so that/3 E F(H0, X, b) = F(H0, X, b'). As in Definition 3.1.2 we see

(3.23) Lemma The functor 0 is full and representative. (3.2.4) Remark Let F be a bype functor on C and let a: K - C be a functor. Then we obtain similarly as in Remark 3.1.4 the bype functor a *F on K and the induced functors

a: bypes(K,a*F)-> bypes(C, E) a:Bypes(K,a*F)- Bypes(C, F). The induced functors are equivalences of categories if a is.

Now let X = (X, H0, Ht, b, /3) be F-bype as above. Then we obtain elements /3* =µ(/3)EHom(Ho,FOX) (3.2.5) (3t =0-'i*PEExt(ker/3*,cokb) as follows. The homomorphism /3 * is given by /3 in X and by the natural transformation µ in the bottom row of Definition 3.2.2 (*). By the assumption in Definition 3.2.2 (1) the homomorphism /3* is surjective. Now let

is ker( /3*) = Ho - Ho (1) be the inclusion of the kernel of 6*. Then i yields by naturality a commutative diagram

Ext(H0,cokb) 4* F(HO,X,b) - Hom(H0,FOX) 1;. l,. µ l,. (2) Ext(HH,cokb) No F(Ho,X,b) -. Hom(Ho,F0X) 3 CLASSIFICATION OF HOMOTOPY TYPES 89

Here i *iL( f3) = 0 implies that the element /3t = 0 - I i *( 13) in (3.2.5) is well defined. Let cokb> rr(Ot)-.ker(p*) (3) be an extension of abelian groups which represents the element fat. Hence we obtain by X the exact sequence b ).FIX--i# h . HI - ir( f3t)-*Ho9->FOX-->0 (4) which we call the F-sequence of X. Here i is the composite of the quotient map and the inclusion in (3), and h is given by the projection in (3). The sequence is natural with respect to morphisms between F-bypes in the following sense. We say that a morphism f: X --> X' in C, together with a commutative diagram in Ab, H, - FIX -7T(Pt) _ Ho -* FOX --' 0

if. If. (5) 1 1 He --,FIX' -,r(f3t) -Ho- FOX'- 0 isa weak morphism between F-sequences if there is a homomorphism ir( f3t) - ir( f3) which extends the diagram commutatively. Similarly we define weak isomorphisms. A morphism (f, cpo, cp) between F-bypes as in Definition 3.2.2 clearly induces a weak morphism between f-sequences for which the vertical arrows in (5) are cp F, f, cpo, and Fo f, respectively.

3.3 Duality of bype and kype

Kype functors (E, E0, EI) form an abelian group kextc(E,, Eo) and similarly bype functors (F, F0, F,) form an abelian group bextc(F0, F,). We show for F0 = E0 and F, = EI that there is a duality isomorphism D: kextc(E Eo) = bextc(E0, EI). Hence a kype functor E determines up to equivalence a bype functor F = D(E) and vice versa. We say in this case that E and F are dual to each other, see (3.3.6). Given n e Z and a functor K* : C - Chain1/= we obtain induced kype and bype functors E and F, respectively, with E(X,A)=H^+'(K*X,A) F(A,X)=H(A,K*X) 90 3 CLASSIFICATION OF HOMOTOPY TYPES for A E Ab and X E C. These turn out to be dual to each other; in particular E is split if and only if F is split, see Theorem 3.3.9. Hence the computation of the functor F also yields a computation of the dual functor E. This fact is of crucial importance in our main result on the classification of homotopy types in the next section. Let C be a category. (We assume also that C is a small category so that the cohomology of C is an abelian group. There is, however, a canonical way to extend the following results to the case when C is not small.) Let

(3.3.1) E,, E,: C -* Ab be two functors. We now consider kype functors E and bype functors F which are both associated with the functors E0 = F0 and E, = F,.

(3.3.2) Definition A kype extension E of E, by E0 is a semisplit kype functor E with structure (E0, E,, A, A), that is, E is embedded in the binatural short exact sequence

Ext(EoX,Tr) 11E(X,1r)-Hom(E,X,7r) with X E C, 7T (=- Ab. Two such extensions E, E' are equivalent if there is a binatural isomorphism r: E(X,?r) -E'(X, ir) with Ar= µ and TA = A. Let kextc(E,,E0) be the set of all equivalence classes of such kype extensions. Dually we define:

(3.3.3) Definition A bype extension E of E0 by E, is a bype functor E with structure (E0, E,, A, µ) that is, E is embedded in the binatural short exact sequence

Ext(H,E,X)° E(H,X)-µ Hom(H,EOX) with X E C, H E Ab. Two such extensions E, E' are equivalent if there is a binatural isomorphism T: E(X, 70 __ E'(X, 7r) with µT= µ and TA = A. Let

bextc(Eo,E1) be the set of such bype extensions.

We introduce for E0, E, in (3.3.1) the bifunctor

(3.3.4) Ext(E0, E,): COP X C - Ab which carries the pair (X, Y) of objects in C to the abelian group 3 CLASSIFICATION OF HOMOTOPY TYPES 91

Ext(E(,X, E,Y). Hence Ext(E0, E,) is a C-bimodule which we use as coefficients in the cohomology groups H *(C, Ext(E°, E, )); compare Definition 1.1.15.

(33.5) TheoremThere are canonical bijections kextc(E,, E°) =H'(C,Ext(E°,E,))=bextc(E°,E,) which carry the split extension to the zero-element in H' (C, Ext(E°, E,)),

The bijections in Theorem 3.3.5 yield an abelian group structure for the sets kextc(E,, E°) and bextc(E°, E,) such that we obtain an isomorphism

(3.3.6) D: kext c (E,,E°) = bext c (E°,E,) which we call the duality isomorphism. We say that a semisplit kype functor E isdual to the semisplit bype functor F ifE0 = F0, E, = F,, and D{E} = {F). Here (E) and {F} denote the corresponding equivalence classes. Indeed using Theorem 3.3.5, each E yields up to equivalence a dual F and vice versa, showing that kype functors and bype functors are actually equivalent to each other. We point out that the bijections in Theorem 3.3.5 are reminiscent of Corollary 3.11 in Jibladze and Pirashvili [CA]. The elements in the first cohomology H'(C, D) of the category C are represented by derivations as follows.

(3.3.7) DefinitionLet C be a (small) category and let D be a natural system on C, for example D = Ext(E°, E,) that is D(f) = Ext(E0 X, E,Y) for f: X Y E C. A derivation

d: C -D (1) is a function which associates with each f: X - Y E Can element d(f) E D(f ) such that for composites gf in C we have the derivation formula d(gf) =g*d(f) +f*d(g) (2) where g* and f * are defined by D. Suppose that there exists a function a E F°(C, D), which carries an object X (=- C to an element a(X) E D(lx), such that d(f) = f* a(X) -f *a(Y). (3) Then d is called an inner derivation induced by a; we also write d = da in this case. Let Der(C, D) and Ider(C, D) be the abelian groups of derivations and inner derivations respectively. Then we obtain the canonical isomorphism H'(C, D) = Der(C, D)/Ider(C, D) (4) 92 3 CLASSIFICATION OF HOMOTOPY TYPES where the right-hand side is the quotient group, compare (IV.7.6) in Baues [AH].

Proof of Theorem 3.3.5 We define functions

H'(C,Ext(E°,E,)) (1) Tk as follows. Let E be a kype extension of E, by E0 and choose splittings sX as in Definition 3.1.1 (4). For f: X - YE C we consider the diagram f' E(X,E,X) (Elf)` E(X,ElY) E(Y,EIY)

I,- IIX I'll (2) Hom(E,X,ElX) Hom(E,X,Ely) Hom(EIY,E,Y) (Elf)* (Elf). where the left-hand square commutes. We now get a derivation ds: C -b Ext(E°, E,) by

ds(f)=0-'((Elf)*sX(1EX)-f*sv(1EY)). (3)

For a different choice s'of splitting functions we obtain a EF°(C,Ext(E°,E,)) by

a(X) =A -'(SX(1EX) -s.(1E X)) (4) and we immediately see that ds - ds. = d,,is the inner derivation induced by a. Moreover, if T: E -> E' is an equivalence we define splitting functions sX for E' by s' = TSX so that in this case ds = ds.. This shows that the function Tk in (1) is well defined by

Tk(E} = {ds}. (5)

Next we construct the function Tk in (1). For derivation d: C -p Ext(E0, E,) we define a kype functor Ed with structure (E0, E1, 0, µ) as follows. For XEC,TrEAblet

Ed(X, Tr) = Ext(E° X,,7r) ® Hom(E1X, 7r) (6) with A = inclusion and µ = projection. For g: Tr - 1r' E Ab we define g* =Ed(X,g)=Ext(E°X,g)ED Hom(E1X,g). 3 CLASSIFICATION OF HOMOTOPY TYPES 93

Moreover for f: X - Y E C we define

f * = Ed (f, 1r): Ed (Y, Tr) --, Ed (X, 7T) by the conditions z Ext(E°f, 7T)=f*0 µf*=Hom(E,f,ir) and by the following condition. Let sx: Hom(E1X, a) -+ Ed(X, 7r) be the inclusion of the second summand in (6). Then we set, for (p E Hom(E1Y, 1r), f*sy(cp) =sx(cpE,(f )) - 0(ww*d(f )). (7) The derivation formula for d shows that Ed is a well-defined kype functor with structure (E0, E,, 0, µ). For a E F°(C, Ext(E0, E,)) we construct below an equivalence of kype extensions

0,,: EdEd,1 (8) This shows that the function Tk in (1) given by

Tk{d} = {Ed} (9) is well defined. One readily checks that Tkis the inverse of r .The equivalence Oa is determined by the conditions µoa = µ, 00a = 0, and by

(10)

forcp e Hom(E, X, v). We clearly have g,, ¢a = 0,,g.. Moreover we get f *#a =Y'af *, with f * induced by Ed,,., by the following equations where w E Hom(E1Y, 7r). f*casy((p) =f *(sy((p) - 0cp*a(Y))

=f *sy(cP) - A f *cp* a(Y) =sx(cPE,(f))-i(cp*(d+da)(f))-Of*cp*a(Y) =sx(cpE,(f))+0(cp*[-d(f)-aa(f)-f*a(Y)}) =sx(gPE,(f)) +Op*[-d(f) -f*a(X)} Oaf *sy(0 = 46a(sx(cpE,(f)) - A(cP*d(f ))) = OcP* d(f )

=sx(cpE,(f))-0(cp*(E,f)*a(X))-Ocp*d(f) 94 3 CLASSIFICATION OF HOMOTOPY TYPES

This completes the proof that Tkin (1) is a bijection. In a completely analogous fashion we obtain the bijection Tb: bextc(EO, E,) - H'(C,Ext(EO, E,)). (11) For this let F be a bype extension of E0 by E, and choose a splitting sX as in Definition 3.2.1 (4). For f: X -o YE C consider the diagram (Eof). F(EOX, X) f. F(E0X,Y) F(E0Y,Y) SX Hom(EoX,EoX) Hom(E0X,EOY):(Eof), Hom(EOY,EOY) (Euf).` where the right-hand square commutes. We obtain a derivation ds: C -, Ext(E0, E) by

ds(f)=I -f*sX(lEOX)+(Eof)*sy(lEQY)). (12) The bijection Tb is now defined by Tb{F} = (ds). (13) We leave it to the reader to show that Tb is well defined and a bijection.

The following definition yields many examples of kype functors and bype functors respectively.

(3.3.8) DefinitionRecall that Chains denotes the category of chain complexes (C* , d) of abelian groups. We say that (C* , d) is a free chain complex if all chain groups C,,, n E Z, are free abelian. Now let C be a category and let K* : C --+ Chains/= be a functor which carries each object X E C to a free chain complex K * M. Using cohomology and pseudo-homology we define, for X E C, AEAb, E(X,A)=Hn+'(K*X,A),

Then E is a kype functor and F is a bype functor with the structure given by the universal coefficient sequences

Ext(H,, K*X,A)H Hn+'(K*X,A) -P Hom(H,, +IK*X,A)

K,, X). 3 CLASSIFICATION OF HOMOTOPY TYPES 95

Here the functors E0 = F0 and E, = F, are given by the homology groups E0(X) = H"(K* X), E,(X) = H"+1(K* X).

(33.9) TheoremThe kype functor E with E(X, A) =H"- `(K,, X, A) and the hype functor F with F(A, X) = H"(A, K * X) are dual to each other in the sense of (3.3.6). In particular, E is split if and only if F is split.

Proof For the chain complex K,, = K*X let Z" = kernel(d: K" --* K"-1) and B" = imaged: K"+, --> K") so that the homology H" = H" K,* X has a short free resolution

B"NZ" - H". (1) In fact, since we assume K. X to be free we see that also B" and Z" are free abelian groups. Moreover, the short exact sequence

H,I K"+I/B"+i B" (2) admits a splitting sX of d and a retraction rX of j since B" is free abelian. Putting (1) and (2) together we obtain the exact sequence

- - + (3) Let A be an abelian group with short free resolution A,>A0 --A and let s"dA be the corresponding chain complex concentrated in degrees n and n + 1. Moreover let sA be in the chain complex concentrated in degree n with (s"A)" =A. Then we obtain E(X,A) =H"+(K*X,A)K,kX,s"+'A]=[dx,sA] with [dx, sA,] = Hom(Z", A). On the other hand we get F(A, X) = H"(A, K. X) = [s"dA, K,* X1 = [dA, dx ]. Here dX is the chain complex concentrated in degrees 0, 1 given by dx in (3). Moreover [-, -] denotes the group of homotopy classes of chain maps. We now define a splitting function resp.s, :Hom(A,H")-1F(A,X) 96 3 CLASSIFICATION OF HOMOTOPY TYPES by use of rx, resp. sx, above. We set, for cp: H+ 1 -A,

E [dxISA]. (4) Since i in (1) is a short free resolution of H we obtain the isomorphism Hom(A, [dA,i] which carries the homomorphism cp to the chain map (cpo, (p1). Then sY is given by

sx, (0 = {(sx'0, p l)) E [dA,dx]. (5) One readily checks that sX and sX are both natural in A. We now consider the derivation d: C -p Ext(E0, E1) given by sX, resp. sX; compare Theorem 3.3.5 (3). Let f: X -* YE C and let f*: K*X=K* ->K*Y=K*' be a chain map representing K,, (f ). We obtain by (1) the equation

Ext(H,,,H, 1)=Hom(B,,,H; (6) Moreover we obtain the following diagrams rxHl =Hn+1K*X lI.1 I(f.1).

Kn+1/Bn+1 ry-H,+1 =Hn+IK*Y

sX B- Kn+1/Bn+1

-s *

These diagrams need not commute. They define, however, factorizations df, resp. 8f, given by

a (Elf )rx-rrf,,+1{' =dfd: (7)

-f,lsx+sYf,, i Kn+1/Bn+1. (8) One readily checks that the derivation ds defined in Theorem 3.3.5 (3) by s =SEsatisfies ds(f) = {df). (9) 3 CLASSIFICATION OF HOMOTOPY TYPES 97

Similarly the derivation ds defined in Theorem 3.3.5 (12) by s= sX satisfies

ds(f) _ (Sf). (10) The right-hand sides of (9) and (10) denote the cosets in (6) given by (7) and (8) respectively. We claim that rx can be chosen via sx such that df= Sf. This implies that E and F are dual and the proof of Theorem 3.3.9 is complete. In fact, we can define rx above by the formula

rx(z)=j-'(z-sxd(z)) (11) where we use j and d in (2), z E Now (11) implies df = Sf since we get the following equations. j dfd(z) =j((E, f)rx(z) -rrfn ,(z)) =jE,(f)j-'(z-sxd(z))-jj-'(frt

=fr+,(z) +syf,, d(z) =j 8fd(z).

(33.10) RemarkIf E and F are dual to each other there should be a connection between the corresponding categories of E-kypes and F-bypes, respectively, which we do not know in general. The detecting functors A, A' in the classification theorem of the next section, for example, yield such a connection. Also if E and F are both split we describe in Section 3.6 the relation between E-kypes and F-bypes.

3.4 The classification theorem We describe in this section fundamental results on the classification of homotopy types by using kype functors and bype functors as defined in Sections 3.1 and 3.2. These results are the main motivation to study suchfunctors.Recallthattypes,,denotesthefullhomotopy category of (m - 1)-connected (m + r)-types and that spaces, denotes the full homotopy category of (m - 1)-connected CW-spaces X with dim X- m + r. For any full subcategory C c typesM'- Ilet types(C) c typeset , spaces' '(C) c spacesM ' be the full subcategories of objects X for which the (n - 1)-type P -,X is an 98 3 CLASSIFICATION OF HOMOTOPY TYPES object in C, n = m + r.If C is the whole category typeset 'then the inclusions in (3.4.1) are equations; in this case we can omit C in the notation. We always assume that m >_ 2. We now introduce a kype functor E and a bype functor F on C with the property E0 = F0 and E, = Fl. We obtain the kype functor

(3.4.2) E: COP x Ab --> Ab by the cohomology group (n = m + r)

E(X,ir)=H"(X,zr) (1) for X E C, 7r E Ab. The kype structure of E is given by the universal coefficient sequence

NH"+'(X,ir) H X X = H X are the homology groups. On the other hand, we define the bype functor

(3.4.3) in two ways, by the functor F;;_ , orby the pseudo-homology,

F(H, X) = F,,'- IM, X) (1) (2) with H E Ab and X E C.RecallthatF,,'- ,(X) = ker (i_ ,X) = image (b X) c r._ ,X leads to the definition of F,"-,(H, X) by the pull-back diagram, see Definition 2.2.9,

(3) N F;;_,(H,X) -.

Moreover we have by Theorem 2.6.14 (5) the commutative diagram

H (4) II 6, 110 11 e. No F;;_(H,X) µ-Hom(H,I'_,X) 3 CLASSIFICATION OF HOMOTOPY TYPES 99

Here O is a natural isomorphism. The short exact sequences of this diagram describe the hype structure of the bype functor F with

FOX=F;;_,X=H"X=EOX, (5) (6)

Since X E C is a 1-connected (n - 1)-type the isomorphisms in (5) and (6) are given by the secondary boundary

F X, b,, X: H"XaI;;_,XcF,X (7) which we use as identification.

Remark The kype functor E in (3.4.2) and the bype functor F in (3.4.3) are examples of functors as defined in Definition 3.3.8. In fact, for C c types, ' we have the functor

K,,: C --> Chain_,/= which carries an object X in C to its singular chain complex. Then clearly

E(X,Tr) =H"+'(K,*X,ir) =Hn+'(X,Tr) F(H,X)=H"(H,K,,X)=H"(H,X).

Hence Theorem 3.3.9 implies that E and F are dual to each other, in particular E is split if and only if F is split. In fact, the duality can be used for the computation of the functor E as follows. First compute I;, _ (H, X) in (3.4.3) (3) and then form the pull-back r,;_,(H, X) which, by (3.4.3) (2), yields F(H, X) together with its functorial properties. Next use duality to derive from F the functor E. We shall apply this method in various examples.

We are now ready to state our main general result on the classification of homotopy types. For this recall that a detecting functor A: A - B reflects isomorphisms, is full, and representative. In particular A induces a 1-1 correspondence between isomorphism classes of objects in A and isomorphism classes of objects in B. A `good classification theorem' inalgebraic topology can often be stated by saying that a certain functor A: A - Bis a detecting functor. Here B is supposed to be a category which is appropriate for computation and A is a more intricate topological category. The next result is crucial for this book; it was announced in Baues [HT]. 100 3 CLASSIFICATION OF HOMOTOPY TYPES

(3.4.4) Classification theoremLet m > 2 and let C be a full subcategory of types'' ' and let E and F be defined as in (3.4.2) and (3.4.3). Then there are detecting functors A: spaces' '(C) --> Kypes(C, E) A': spaces' '(C) -> Bypes(C, F). Moreover the F-sequences of both A(X) and A'(X) with X E spaces, ' (C) are naturally weakly isomorphic to the part

F,,X-> 0 of Whitehead's exact sequence, n = m + r. In addition one has detecting functors A: types,,(C) - kypes(C, E) = Gro(E), A': types,,(C) -> bypes(C, F).

We give an explicit definition of the detecting functors as follows.

(3.4.5) DefinitionRecall that m > 2 and n = m + r. For the definition of A and A we use the Postnikov invariants k, X and for the definition of A' and A' we use the boundary invariants &X. Let

X E spaces' '(C). (1)

Then the (n - 1)-type P_ ,X of X is an object in C and we set

A'(X) = (P,X, H, X, H»+ IX, X, Next let

XE types (C). (2) Then we again have P,,X E C and we set

A(X) _ see(1.3) A'(X) _ Since X is a 1-connected n-type we see that F, XcF.X=F,(X) is injective. 3 CLASSIFICATION OF HOMOTOPY TYPES 101

On the proof of Theorem 3.4.4Only the detecting functor

A: Gro(E)

is a classical result, due to Postnikov. The other detecting functors A, A', A' have not appeared in the literature. We prove that A, ,k', and A are detecting functors in Section 3.7. The proof that A' is a detecting functor is highly sophisticated; it involves most of the theory in the chapter of CW-towers, see Section 4.7.

The functors of the theorem are part of the following diagram which commutes up to canonical natural isomorphisms.

Kypes(C, E) *-"- spaces;, '(C) - - Bypes(C, F)

(3.4.6) kypes(C, E) types,,(C) bypes(C, F)

Here Pm+r is the Postnikov functor. All functors in the diagram are full and representative. But 0 and Pm+r are not detecting functors since they do not reflect isomorphisms. We deduce from Theorem 3.4.4 the next result on the realizability of r-sequences. Recall that an n-equivalence X - Y is a map which induces isomorphisms 7r; X = 7r;Y for i < n.

(3.4.7) Theorem on the realizability of the Hurewicz homomorphismLet Y be a simply connected (n - 1)-type and let

Hlb >FY it ->Ho- be an exact sequence of abelian groups where H, is free abelian and where and are given by Y. Then thereexists an (n + 1)-dimensional CW-complex X, an (n - 1)-equivalence p: X - Y, and a commutative diagram

H,,,, X -- F,, X -- 7r X -- H X -- ker(i- iX) - 0 P. =10 P. H, ---> -b 7r Ho ---a -p 0 in which all vertical arrows are isomorphism.

We can prove Theorem 3.4.7 either by the detecting factor A or by the detecting functor A' in Theorem 3.4.4. In the following proof we use A'. 102 3 CLASSIFICATION OF HOMOTOPY TYPES Proof Let the bype functor F be defined as in Theorem 3.4.4 with C = types;, r + 2 = n - 1. Then we find an F-bype

Here Ho and HI are given by the exact sequence and K is the kernel of 7r and b is given by b1. Let (7r) be the element

{7r} E Ext(ker(bo), FRY/K) determined by the extension fRY/K N 7r -* ker(bo) which we deduce from the exact sequence in the theorem. Then there exists an element 8 with

(3* = boand/3t = {7r},

This follows since µ in (3.2.5) is surjective and since for the inclusion is ker(bo) cHo also the induced map i*:Ext(H0, FIX/K) --> Ext(ker(bo), F1X/K) in (3.2.5) (2) is surjective. Hence the r-sequence of Y is weakly isomorphic to the exact sequence (3.4.7). Now let X be a A'-realization of Y. Thus we obtain the proposition by the natural isomorphism of r-sequences in Theorem 3.4.4.

(3.4.8) Remark Theorem 3.4.7 shows that each exact sequence H, is realizable. We do not know however what morphisms between such sequences are realizable. More precisely let X, X' be 1-connected (n + 1)- dimensional CW-complexes with (n - 1)-types Y = P_ ,X and Y' = P_ 1X' respectively. Then we consider the commutative diagram in Ab

H, --b 1'Y ---> 7r ---> Ho -i ker i_IY ----> 0

11P, If. I90 I'o If. (1) H,' -- F,, Y' --> 7r' - Ho - ker i _,Y' ----> 0 where f E [Y, Y'] and where the top row and the bottom row are Whitehead's exact sequence for X and X' respectively. The detecting functor A shows that (f, co, q'1) is realizable by a map f : X -* X' if and only if

f (.P,r) * (2) On the other hand, the detecting functor A' in Theorem 3.4.4 shows that (f, cpo, (p1) is realizable by a map X -> X' if and only if

(3) 3 CLASSIFICATION OF HOMOTOPY TYPES 103

What is the condition that (f, cpo, p1, is realizable by a map X -'X'? Clearly (2) and (3) must be satisfied but is this a sufficient condition? Moreover what pairs of invariants are realizable? Hence we are searching for an unknown category U for which the diagram of detecting functors

spacesm 1(C) A A' (4)

Kypes(C, E) F- U ---a Bypes(C, E) commutes and for which U is given by an algebraic structure like E, F on C. This is the unification problem. Since A and A' are detecting functors we see that they induce a 1-1 correspondence between isomorphism classes of objects in Kypes(C, E) and Bypes(C, F) respectively. In this sense the k-invariant k"(X) determines the boundary invariant 9"(X) and vice versa, but it is unclear how this connection between kn(X) and 9"(X) could be described algebraically. Below we show that k, ,X is `orthogonal' to 8,(X). Moreover, in the case that E and F are split, we describe a possible candidate for the category U in Definition 3.6.1 (6), namely U = S(E0, E,).

(3.4.9) DefinitionLet X be a space and let C. X be the singular chain complex of X. Then we know that the cohomology and pseudo-homology can be described by sets of homotopy classes of chain maps Hn+ 1(X, A) _ [C,k X, C, M(A, n + 1)] H"(B,X) _ [C.M(B,n),C.X]. Hence we get by composition of chain maps a pairing H"(B, X) ®Hn+I(X, A) --> Ext(B, A) which carries /3 (9 k to ( )3, k) = k ° /3 in [C,, M(B, n), C,, M(A, n + 1)] = Ext(B, A).

(3.4.10) PropositionLet X be a simply connected CW-space with (n - 1)-type pn_,: X->Y=Prs_1Xand let knXEHn+1(Y,-rr"X),resp./i"XEH"(H"X, Y)/A image (p,,-,) be the k-invariant, resp. boundary invariants, of X. Then k" X and /3,, X are orthogonal with respect to the pairing (,) in Definition 3.4.9, that is (/3,k"X)=0 for all / E H"(H" X, Y) representing /3" X. 104 3 CLASSIFICATION OF HOMOTOPY TYPES

Proof Let H = H X and 7r = ir X and consider the composite

k k (X ). The Postnikov tower shows that k p_ I -0 is null homotopic. We now obtain the following commutative diagram

H H H H (HK11 n + 1)) H, n OTn Since /3 E 8,, X is in the image of b and (p_I)* respectively, we see that /3=0 since k p_ , -0. Here /3 represents (,B, k X ).

3.5 The semitrivial case of the classification theorem and Whitehead's classification

In this section we describe the homotopy classification of (m - 1)-connected (m + 2)-dimensional CW-spaces by simple algebraic invariants. This corresponds to well-known results of J.H.C. Whitehead [CE], [SC], [HT]. We obtain the classification by applying the classification theorem 3.4.4 to the simple case r = 1;in fact, for r = 1 only semitrivial kype functors and semitrivial bype functors are relevant. Recall that for an Eilenberg- Mac Lane space K(A, m), m >: 2, one has natural isomorphisms

Hm+,K(A, m) = Fm+IK(A, m) = (3.5.1) Hm+IK(A,m) = rmK(A,m) = 0.

Here Ab -> Ab is the algebraic functor with r2(A) = r(A) (given by Whitehead's quadratic functor r) and with Fm(A) =A ® Z/2 for m > 3. For an (m - 1)-connected space X we have a natural isomorphism (see Theorem 2.1.22)

(3.5.2) Fm +,X-r,,(HmX) so that Whiteheads sequence yields the exact sequence

(3.5.3) Hm+2X b>rm(HmX)-L1Tm+1X-'Hm+iX 40 which is natural for maps between (m - 1)-connected spaces. Here Hm+2X 3 CLASSIFICATION OF HOMOTOPY TYPES 105

is free abelian if dim X< m + 2. We now introduce the following categories with objects being exact sequences as in (3.5.3).

(35.4) DefinitionLet C be a category and let G: C --> Ab be a functor. A G-sequence is an object H in C together with an exact sequence of abelian groups H, -U G(H) --7r --+ Ho -* 0. (1) A morphism between G-sequences is a morphism cp: H --> H' in C together with a commutative diagram

H, ---> G (H) -- p7r ---> Ho ---> 0 IW. (2) I 1,,- H, --b G(H') ---b 7r' -a Ho -0

A weak morphism between G-sequences is a triple (cp,,Po, cp) for which there exists 4 such that the diagram commutes. The G-sequence (1) is free if H, is free abelian and is injective if H, - G(H) is injective. Let K(G),resp.k(G) (3) be the categories consisting of free,resp.injectiveG-sequences and morphisms as above. Clearly k(G) = Gro(Hom(G, - )) is the Grothendieck construction of the bifunctor Hom(G,-): C°P X Ab -->Ab which carries (H, a) to the abelian group of homomorphisms Hom(G(H),ir). Moreover let B(G),resp.b(G) (4) be the categories consisting of free, resp. injective G-sequences and weak morphisms. The proof of the next lemma is left as an exercise.

(3.5.5) Lemma Let E be a semitrivial kype functor with E0 = 0 and E, = G. Then one has canonical equivalence of categories Kypes(C, E) = K(G), kypes(C, E) = k(G). Let F be a semitrivial bype functor with F. = 0 and F, = G. Then one has canonical equivalences of categories Bypes(C, F) = B(G), bypes(C, F) = b(G). 106 3 CLASSIFICATION OF HOMOTOPY TYPES

In the next result we consider the case when G = F,,,: Ab -* Ab is the functor given by (3.5.1). Recall that spacesm is the full homotopy category of (m - 1)-connected (m + 2)-dimensional CW-spaces.

(3.5.6) Theorem of J.H.C. WhiteheadFor m > 2 one has detecting functors A: spaces' -

A': spacesm B(F,n)

These functors carry a space X to the exact sequence in (3.5.3).

Proof Consider Theorem 3.4.4 for the special case r = 1 and C =typeset. Then one has an equivalence of categories C = Ab and by (3.4.2) and (3.5.1) the kype functor E on Ab is given by

E(A,Tu) = ir) (1) with E0 = 0 and El = F,,,. Hence one has, by Lemma 3.5.5, an equivalence of categories Kypes(C, E) = K(F,,) (2) and the detecting functor A in Definition 3.4.5 corresponds to the functor A in Lemma 3.5.5. We now consider the detecting functor A' in Theorem 3.4.4 for the special case r = 1 with C as above. Then we have by (3.5.1) and (3.4.3) the hype functor F on Ab given by

F(H, A) = Ext(H, F,,,(A)) (3) with F0 = 0 and FI = Moreover we have by Lemma 3.5.5 an equivalence of categories Bypes(C, F) = B(r,',). (4) Thus the detecting functor A' in Definition 3.4.5 yields the functor A' in Lemma 3.5.5. 0

(3.5.7) Remark One has a forgetful functor ': K(G) -, B(G) such that for G = F,,, the functors in Lemma 3.5.5 satisfy 4A = A'. Since 0 and A are both detecting functors this also shows that A' is a detecting functor. 3 CLASSIFICATION OF HOMOTOPY TYPES 107

Recall that types' is the full homotopy category of (m- 1)-connected (m + 1)-types.

(3.5.8) TheoremFor m >- 2 one has detecting functors A: types' - k(F ;) = A': types' -,, b(F,;,).

These functors carry X E types' to the exact sequence (3.5.3).

Proof For the kype functor E in Theorem 3.5.6 (1) we have by Lemma 3.5.5 kypes(C,E) = k(I ;) and for the bype functor F in Theorem 3.5.6 (3) we have an equivalence of categories bypes(C, F) = b(I',' )

Hence Theorem 3.5.9 is a consequence of the classification result Theorem 3.4.4.

3.6 The split case of the classification theorem

We here discuss the classification theorem 3.4.4 in case the bype functor F and the kype functor E are split with E0 = F0, E, = F,. In this case E-kypes and F-bypes can both be described by chain complexes H, -->E1(X) ->R ->E0(X) - 0 which are exact at E,(X) and E0(X). This simplifies the description of the corresponding categories of E-kypes and F-bypes considerably. Hence for split functors E, F we get a new kind of classification theorem derived from Theorem 3.4.4. As an illustration we consider the homotopy classification of (m - 1)-connected (m + 3)-dimensional polyhedra X with trivial homotopy groups IT,,, + , X = 0, m > 4. In this case the bype and kype functors are split and are of a particularly easy form. Various other examples of split hype and kype functors are described in Chapter 6.

(3.6.1) DefinitionLet C be a category and let E0, E,: C - Ab be two functors. An (E0, E, )-sequence is an object X in C together with a chain complex H,-b-)I E,(X) -1), R-84EO(X) -o 0 (1) 108 3 CLASSIFICATION OF HOMOTOPY TYPES of abelian groups which is exact in E,(X) and E0(X). A morphism between such sequences is given by a morphism f: X --'X' in C and by a commutative diagram in Ab HI - E1(X) R 5 Eo(X)

IW1 If. Ir if. (2) H; -b--a EI(X') L R' -s-> E0(X')

The (E0, El)-sequences and such morphisms form a well-defined category. We say that the sequence (1) is free if H, is free abelian and we say that (1) is injective if b is injective. Let

S(E0, EI ),resp.s(E0, EI) (3) be the full subcategories of free, resp. injective (E0, E,)-sequences. We k b introduce two natural equivalence relations and as follows. Let k ((pl, f, r) and (,f', r') be morphisms as in (2). We set ((p f, r) (,f', r') if cp, = Bpi, f =f' and if r and r' induce the same homomorphism.

r* =r*': kernel(8) --> kernel(S'). (4)

b On the other hand, we set (cp,, f , r)(V f', r') if cpl = cp,, f = f' and if rand r' induce the same homomorphism

r* = r*': cokernel(d) -> cokernel(d'). (5) One has the obvious quotient functors

S(E0,E,)/b 4-S(E0,EI)-->S(E0,E,)/k (6)

s(E0,EI)/b F-s(E0,El)--.s(E0,E,)/k (7) all of which are easily seen to be detecting functors, in fact linear extensions of categories. We associate with an object (1) the following exact sequence

HI - E,(X) -'-a ker(b) a cok(d)b0.Eo(X) -->0

(8)

Iff Ho

Here bo is induced by S and i is induced by d. Moreover j is the composite j : ker(5) c R -. cok(d ) 3 CLASSIFICATION OF HOMOTOPY TYPES 109 of the inclusion and the quotient map. Morphisms between(E0, E,)-sequences clearly induce morphisms between the corresponding exact sequences in (8) which we call I'-sequences.

(3.6.2) Lemma Let E be a split kype functorgiven by E0, E, and let F be a split hype functor given by F0, F,; see Definitions 3.1.1 and 3.1.2. Then there are canonical equivalences of categories

-r: S(Eo,E,)1!-> Kypes(C, E)

,r: s(E0, E,)/k-+ kypes(C, E)

T':S(Fo,F,)/b -Bypes(C,F)

T': s(F0, F,)/ b -+ bypes(C, F).

Proof The equivalence T carries the (Eo, E, )-sequence S={H, 2-41 E,(X)__d;IR8Eo(X)--,0} to the kype T(S)=(X,ir,k,H,,b)withor=ker(b) where k E E(X, ir) = Ext(E0X, 7r) x Hom(E,X, ir) is given by is E,(X) ---' ?r in Definition 3.6.1 (8) and by the extension 0-+1r-*R-->EOX->0 given by 3. Now one readily checks that T in the statement of the lemma is an equivalence of categories. Next let S' be the (F0,F,)-sequence S' _ {H, -F,(X) _24 R '+F0(X) -+0}. Then T' carries S' to the bype withHo=cok(d). Here $ E F(H0, X, b) = Ext(Ho, cok b) x Hom(H0, FOX) is given by bo: Ho - FOX as in Definition 3.6.1 (8) and by the extension 0-'cok(b)L,R- Ho -'0 where d' is induced by d. Again one readily checks that T' in the statement of the lemma is an equivalence of categories. 110 3 CLASSIFICATION OF HOMOTOPY TYPES

We are now ready to formulate an addendum to the classification theorem 3.4.4 which deals with the case when the bype and kype functors are split.

(3.6.3) Classification theoremLet m >_ 2 and let C be a full subcategory of types,, ' and let E and F be defined as in (3.4.2) and (3.4.3). Then the kype functor E on C is split if and only if the bype functor F on C is split. If E and F are split we obtain, with the homology functors (n = m + r) Hn,Hn+1:C-*Ab, the following detecting f inctors: A: spaces;,'(C) _ S(Hn, Hn+1)/

A': spaces, '(C) -* S(H,,, H,+ 1)/

k A: types' (C) -> s(Hn, Hn+ 1)/

,V: typesm(C) -> s(Hn,Hn+I)/b Moreover the F-sequences of A(X) or A'(X) given by Definition 3.6.1 (8) are natural weakly isomorphic to the part

Hn+1X-FnX--* of Whitehead'sexact sequences.Here we useF,, X = Hn+ ,Pn-,X and ker(in_,X)=H,, P._1X. Proof We apply the remark below (3.4.3X7). Hence the theorem is a consequence of Lemma 3.6.2 and the classification theorem 3.4.4. Remark We do not see that there is a functor spaces; 1(C) 2S(E0,E1) which induces the functors A and A'. Theorem 3.6.3, however, suggests that there might be such a functor in case E and F are split.

We now consider an example for the classification theorem 3.6.3. For any abelian group A we have the exact sequence (3.6.4) 0-A*l/2->A- .A-A®71/2--*0 where A *Z/2 is the 2-torsion of A and where A/2A =A ® 1/2 is the tensor product of A with 71/2. Thus we obtain functors *71/2, ®71/2: Ab -i Ab which carry A to A * 71/2 and A 0 71/2 respectively. The next result is an application of the classification theorem 3.6.3. Let spaces(m, m + 2). c spaces, 3 CLASSIFICATION OF HOMOTOPY TYPES 111 be the full homotopy category of (m - 1)-connected (m + 3)-dimensional CW-spaces X with 7Tm+ 1 X = 0.

(3.6.5) TheoremLet m > 4. Then there are detecting functors

spaces(m, m + 2) - We ZL/2, * Z/2)/k

spaces(m,m + 2)-"* S(®71/2, *71/2)/ Moreover the F-sequences of A(X) and A'(X) with A = ITm X = HX are natural weakly isomorphic to the part

Hm+3X* rm+2X- 7rm+2X -'Hm+2`Y-'rm+1X--* 0 of Whitehead's exact sequence.

Theorem 3.6.5 implies by Definition 3.6.1 (6) that there is a 1-1 correspondence between (m - 1)-connected (m + 3)-dimensional homotopy types X with 7rm+X = 0 and m >- 4 and isomorphism classes of objects 1 H1 -*A*Z/2-R->A®Z/2-*0 in S((&71/2, *Z/2); see Definition 3.6.1. This is indeed a simple description of such homotopy types.

Proof of Theorem 3.6.5 We consider the case r = 2 in the classification theorem 3.6.3 where we set C = types= Ab Eilenberg and Mac Lane [II] show that the kype functor E is split with E0X =Hm+2K(A,m) =A ®Z/2 E1X =Hm+3K(A, m) =A *71/2 for X = K(A, m) E C. Hence, by Theorem 3.3.9, also the bype functor F is split with F0 = E0 and F1 = E1. Below we shall prove that the bype functor F is split independently of Theorem 3.3.9. Now the application of Theorem 3.6.3 completes the proof. 0

3.7 Proof of the classification theorem

We assume that m z 2, n = m + r, and

(3.7.1) C = typeset '. 112 3 CLASSIFICATION OF HOMOTOPY TYPES

The kype functor E and the bype functor F on C are given by E(X, vr) = Hn+ 1(X, 7r) and F(H, X) = Fi_1(H, X) with X E C, 7r, H E Ab; see (3.4.2) and (3.4.3). We first show the classical result of Postnikov:

(3.7.2) PropositionThe functor A: types, , Gro(E) A(X) = (P,-1X,ir,X,k,X) is a detecting functor.

Proof The functor A reflects isomorphisms by the Whitehead theorem. Moreover an object (Y, ir, k) in Gro(E) with k e E(Y, Tr) is A-realizable by choosing X with d(X) = (Y, r, k) as follows. Let X = Pk be the fibre of a map k: Y --> K(vr, n + 1) which represents the cohomology class k. Each morphism (f, cp): A(X) - A(X') is A-realizable since the diagram

ykK(IT, n + 1)

kLK(7r',n+1) Y' homotopy commutes by condition (3) of Remark 3.1.3. Hence there is an associated principal map X = Pk - Pk' = X' which realizes (f, (p), see (V. §6) in Baues [AH].

Next we consider the functor A in Theorem 3.4.4.

(3.7.3) PropositionThe functor A: spacesm 1 -+ Kypes(C, E) A(X) = (Pn+1X,ir,X,Hn+1X,bn+1X) is a detecting functor.

Here we use (Pn_ ,X, 9rn X, kn X) E Gro(E) as in Proposition 3.7.2 and bn+1X: Hn+1X-1',X=Hn+1Pn-1X is given by the secondary boundary operator and the isomorphism 0. By Theorem 2.5.10 (c) we see that bn+ ,surjects to the kernel of p.(kn X) = (k,X), =inX. Hence naturality of k,X shows that the functor A in Proposition 3.7.3 is well defined. 3 CLASSIFICATION OF HOMOTOPY TYPES 113

Proof of Proposition 3.7.3Itis clear by the Whitehead theorem that A reflects isomorphisms. We now show that A' is representative, that is, each kype (Y, 7r, k, H, b) = Y has a A-realization X with A'(X) __ Y. For the construction of X we first choose an n-type U with k(U) = (Y, 7r, k); we can do this by Proposition 3.7.2. Here we may assume that U is a CW-complex. We now construct X together with a map X- U which induces isomorphisms of homotopy groups 7r, for i >_ n so that AX = AU. For the cellular chains C* U and the skeleton U" of U we obtain the following commutative diagram with exact columns

k*. H-ker(kI) c FnY 7r sl r J II Cn+1U'ker(i*)c7r"Un ,,TnUn+l

K cC"U

Here I'"Y= F"U is a subgroup of 7r"U" and µ(k) = k* is a restriction of i* by Theorem 2.5.10 (b). This shows that the quotient K = ker(i *)/ker(k * ) injects into the free abelian group CnU and hence K is free abelian. Therefore we can choose a splitting t of the surjection C"+ 1 U - K given by the attaching map f of (n + 1)-cells in U. We define the CW-complex X = X"+' by the n-skeleton X" = U" and by the attaching map of (n + 1)- cells g: C"+ 1 X -* 7r" X". Here C"+ 1X is the free abelian group H ®K and g is the composite

g: Cn+1X=H®K-*C"+1U111 7r"U" (2) where s: H -> C"+ 1U is any homomorphism for which diagram (1) commutes. Since g =f o(s, t) we obtain a map X -+ U which is the identity on the n-skeleton and which induces (s, t) on cellular chains C,,,,. Since by construction of g image(g) = ker(i *) (3) we get 7r" X = 7r,, U = IT. Moreover the construction of g shows H,,X = H and b"1 X = b so that X in fact is a A-realizationof Y above. It remains to show that the functor A isfull. For this let X, Y be CW-complexes in spacesm 1 and let (f, cp, cpH): A'(X) - A'(Y) (4) be a morphism between kypes. Let U = P" X with U"+' = X"' = X and in the same way let V = P"Y with V"+' = Y"+ 1 = Y. Since (f, cp): A(X) --> A(Y ) 114 3 CLASSIFICATION OF HOMOTOPY TYPES is a map of Gro(E) we obtain by Proposition 3.7.2 a cellular map g': U--> V which realizes (f, q ). Hence the map g' restricted to the (n + 1)-skeleton yields a map g: X -> Y which realizes (f, (p) but which need not realize (pH. We obtain the following diagram, where the left-hand side is defined by X and the right-hand side is defined by Y; compare diagram (1). H wH H'

II (*) II

H,,,rn IX CC,,,,X g

lb iT,,Ifx Ifr ib' r,, g. X C X " - rr"Y" Y

r,g = r,J All subdiagrams commute except possibly subdiagram (*). We have however by Definition 3.1.2 (3) the equation b',PH = (F g)b. (6) The maps fX, fY are the attaching maps of (n + 1)-cells in X and Y respectively. Now (6) and (5) imply that the difference PH-g*: H-->Cn+,Y given by diagram (*) satisfies

fY((pHg*)=O. (7) Since H is free abelian and since the sequence

n+l Trn+1(Yn+1,Yn) 1 ,7r"Yn 7r

11 fy Cn+1X is exact we can choose by (7) a homomorphism

1Yn+l a: H- ir,, withj(a)=coH-g*. (8) Now H is a direct summand of Cn+,X for which we choose a retraction r so that we get a map h=g+ar: X -->Y (9) by the action in (4.2.5). The map h is cellular and satisfies h" =g" on the n-skeleton. We claim that h is in fact a A-realization of (f, gyp, (PH) above. Indeed since h" =g" the map h realizes (f, cp) and we compute Hn+ 1(h) by Cn+1(h)=(g+ar)* =g* +jar=g* +(cpH-g*)r (10) so that the restriction of C"+ 1(h) to H yields pH. Hence h realizes also cpH. Therefore the proof of Proposition 3.7.3 is complete. 0 3 CLASSIFICATION OF HOMOTOPY TYPES 115

Finally we consider the functor A' in the classification theorem 3.4.4. (3.7.4) PropositionThe functor A': types, - bypes(C, F) with A'(X)=(Pn X,HnX,Hn1 X,bn+1X,R»X) is a detecting functor.

We here only show that Proposition 3.7.4 is a corollary of the corresponding result:

(3.7.5) PropositionThe functor A': spaces; 1 -* Bypes(C, F) with A'(X)=(Pn X,bn+1 X,OnX) is a detecting functor.

The highly sophisticated proof of Proposition 3.7.5 involves most of the theory of Chapters 2 and 4. In particular the new concept of towers of categories is crucial for this proof. The final proof of Proposition 3.7.5 is given in Section 4.7. Proof of Proposition 3.7.4 We derive Proposition 3.7.4 from Proposition 3.7.5. Again the Whitehead theorem shows that the functor A' reflects isomorphisms. In fact, recall that /in X determines (7r,,X) in Theorem 2.6.9 (c). Hence the five lemma shows that an isomorphism between bypes yields an isomorphism on zr* X. Next we consider realizability of objects. For this we use the commutative diagram of functors in (3.4.6). In this diagram it is easy to see that each bype T has a 4-realization T'. Since A' is a detecting functor we find a A'-realization T" of T'. Hence P,n+,T" is a A'-realization of T since diagram (3.4.6) commutes. Similarly we see that the functor A' is full: let T1, T2 be objects in types'. and let f: A71 --> A72 be a morphism between bypes. Let T1', T2 E spaces'. be the (m + r + 1)-skeleta of T1 and T2 respectively. Then clearly PT,' = T1 and PT2 = T2. Moreover we can choose a morphism f': AT; - AT2 with Of = f. For this we only choose a homomorphism f' such that the diagram Hn+1T1' - bHn+1T1

Hn+1T2 bHn+1T2 commutes. This is possible since the horizontal arrows are surjective and since H,,, ,T,'is free abelian. Now, since A' is a detecting functor, we find a A'-realization f": T1 - T2 of f and hence Pf" is a A'-realization of f. This completes the proof that A' is a detecting functor. 4 THE CW-TOWER OF CATEGORIES

A CW-complex X is obtained inductively by constructing the skeleta X", n >- 0. Since we here only consider homotopy types of simply connected CW-complexes we may assume that X is reduced in the sense that the 1-skeleton of X consists of a single point, X' = *. However, the skeletal filtration has the disadvantage that the homotopy type of X" is not well defined by the homotopy type of X. For this reason J.H.C. Whitehead introduced the (n - 1)-type of X which is also the (n - 1)-type of X" and which can be obtained from X" by `killing' homotopy groups lrm X", m >- n. The (n - 1)-type is the (n - 1)-section of the Postnikov tower of X which we denote by P,,_1M. It is a classical result that the homotopy type of P"_1(X ) is well defined by the homotopy type of X. This fact justifies the construction of Pn_1(X), though it is to some extent absurd to replace a nice n-dimensional CW-complex X" by a CW-complex P"-1(X) which in general is infinite dimensional, and the homology of which is hard to compute. We here study the skeletal filtration of a CW-complex X rather than the Postnikov decomposition. We deduce from the skeletal filtration the object

rn+ 1(X) = (C,fn+1,X") which is a triple consisting of an algebraic part C (which is the cellular chain complex of X) and of a topological part X" (which is the n-skeleton of X). is the homotopy class of the attaching map of (n + 1)-cells of Moreover fn+1 X given by a homomorphism C,,+ 1 - ir,,X. We call such a triple a homotopy system of order (n + 1). The crucial point is that homotopy systems of order (n + 1) form a homotopy category Hn+ 1/= and that the homotopy type of the object rn±1(X) in Hn±1/= depends only on the homotopy type of X. Hence enriching the n-skeleton X" by an algebraic part (C, fn+ 1) yields a new invariant rrt+1(X) of the homotopy type of X which has the same kind of naturality as the Postnikov section P"_ 1(X). We consider the sequence r3(X),r4(X),...,r"+1(X),... of homotopy systems to be the true Eckmann-Hilton dual of the sequence P2(X), P3(X),..., P"_ 1(X),... of Postnikov sections of the simply connected CW-space X. It is Postnikov's result that the homotopy type of P,,(X) is determined by the pair (Pn_ 1(X),k"(X)) 4 THE CW-TOWER OF CATEGORIES 117 where kn(X) is the nth k-invariant of X. Our main result here shows that, on the other hand, the homotopy type of r,1(X) is determined by the triple (see Theorem 6.6.4) (r (X), f3n(X),bn+1(X)). Here bn+ 1(X) is the second boundary homomorphism in Whitehead's certain exact sequence and $n(X) is the boundary invariant introduced in Chapter 2. Since f3n+ 1(X) determines 1(X) we see that the sequence of boundary invariants /33(X),134(X),... determines the homotopy type of X in a similar way as the sequence of k- invariants

Moreover each simply connected homotopy type can be built by the inductive construction of either sequence. Further dual properties of boundary invariants and k-invariants are described in Chapters 2 and 3. The categories Hn+ 1/" of homotopy systems of order (n + 1) yield the CW-tower of categories. This is a sequence of functors (n >t 3) spaces2 '- Hn+ 1/' x . H3/= which approximates the homotopy category spaces2 of simply connected CW-spaces. Each functor A is embedded in an exact sequence H"f + - Hn+1/= ->Hn/= a H°+1rR. Here Hmfn denotes an Hn/~-bimodule. In Section 4.1 we recall the useful language concerning such exact sequence. (The CW-tower of categories is studied for non-simply connected spaces in Baues [AH] and [CH]. We here deal only with the simply connected case. This simplifies these towers considerably; compare also the final chapter in [AB]). The proofs of our main results are based on properties of the CW-tower of categories. In particular in Section 4.6 we relate the obstruction operator 61 in the CW-tower with the secondary boundary operator of Whitehead and the boundary invariants in Chapter 2. This leads in Section 4.7 to a proof of the classification theorem in Chapter 3. Moreover we prove a theorem on the action H'F,,+ in the CW-tower which is useful for the classification of homotopy classes of maps; see Section 4.8.

4.1 Exact sequences for functors The concept of exact sequences for groups is well known in algebraic 118 4 THE CW-TOWER OF CATEGORIES topology. We can consider a group to be a category with a single object in which all morphisms are equivalences. Therefore there might be a more general notion of an exact sequence for categories and functors. Motivated by the CW-tower of categories we introduce in this section an exact sequence for a functor A of the form D+ -A------4 Here, however, D and H are not categories but natural systems of abelian groups on B, for example B-bimodules. Such natural systems serve as coefficients of cohomology groups H"(B, D) of the (small) category B. Special exact sequences are the linear extensions of B by D. Exact sequences for a functor A and linear extensions arise frequently in algebraic topology and in many other fields of mathematics; see Baues [AH], [CH]. The examples here are mainly derived from the CW-tower of categories. As usual let Ab be the category of abelian groups. For a category B a B-module M is a functor M: B --> Ab and a B-bimodule D is a functor D:B°PxB-->Ab. Here BP is the opposite category and B°P X B is the product category. For objects X, Y in B the set D(X, Y) is an abelian group contravariant in X and covariant in Y. For example if B C Top*/= is a subcategory of the homotopy category of pointed spaces we have the B-bimodule D,

(4.1.1) D(X,Y) = H"(X, 77;,, Y), given by the nth cohomology of X with coefficients in the mth homotopy group of Y. This bimodule arises often in obstruction theory. The following notion of a `natural system of abelian groups on B' generalizes the notion of a B-bimodule. For this recall that the category of factorizations in B, denoted by FB, is given as follows. Objects are morphisms f, g in B and morphisms f -* g are pairs (a, b) for which b 4 Y Y' ff f9 X aX' commutes in B. Thus bfa is a factorization of g. A natural system (of abelian groups) on B is a functor

(4.1.2) D: FB - Ab that is, an FB-module. This functor carries the object f to Df = D(f) and carries (a, b) to D(a, b) = a*b* with a* = D(a,1) and b* = D(1, b). A B- bimodule D is also a natural system by setting Df = D(X, Y) for f: X - Y, 4 THE CW-TOWER OF CATEGORIES 119 that is, in this case Df depends only on the source and the target of f. A functor A: A --> B induces the function

A:A(X,Y) -* B(AX, AY)

between morphism sets; here X and Y are objects in A. For a morphism fo: X - Y in A with f = Afo we thus have the subset

(4.1.3) A-'(f)cA(X,Y) withfoEA-'(f).

Now recall the definition of a linear extension of categories in Definition 1.1.9:

(4.1.4) D+ N A B.

The next definition of an exact sequence for a functor A generalizes the notion of a linear extension in two ways. On the one hand, A needs not to be full but its image can be described by an obstruction operator ®; on the other hand, the action of Df on A-'(f) need not be effective.

(1.4.5) DefinitionLet A: A - B be a functor and let D and H be natural systems of abelian groups on B. We call the sequence D-iA-pB-H an exact sequence for A if the following properties are satisfied.

(a) For each morphism fo: X - Y in A the abelian group Df, f = Afo, acts transitively on the set of morphisms A -'(f ) c A(X, Y). Let I f _ {a e Df, fo + a = fo) be the isotopy group. (b) The linear distributivity law (Definition 1.1.9) (c) is satisfied. (c) For all objects X,Y in A and for all morphisms f: AX -> AY in B an obstruction element ®X. y(f) E H(f) is given such that ®X. y(f) = 0 if and only if there is a morphism fo: X - Y with Afo =f.

(d) ® is a derivation, that is ®X, z(gf) =g * ®X, y(f) + f *®y. z(g)for f : AX - AY, g: Ay- * AZ. (e) For all objects X in A and for all a E H(lAX) there is an object Y in A with AY= AX and ®X,y(l,AX) = a; we write X = Y+ a in this case. 120 4 THE CW-TOWER OF CATEGORIES

A tower of categories is a diagram (i E Z)

I D, --+Hi -'ri+1 (f) I A

D,,-1 -> Hi-1 --'1,

1 where D. - H; -> Hi- 1 -+ f1 is an exact sequence. We say that D acts on A if Definition 4.1.5(a) above is satisfied. Moreover, D acts linearly on A if (a) and (b) are satisfied. We say that D acts effectively if all isotropy groups in (a) are trivial. A linear extension as in (4.1.4) yields an exact sequence D±,E- C0-b where 0 is a trivial natural system. On the other hand, each exact sequence as in Definition 4.1.5 yields a linear extension of categories

(4.1.6) D/INA-r AA. Here AA is the image category of A: A - B. The natural system D/I on AA is given by (D/I )(f) = Df/I fo, fo E A-' (f ); see Definition 4.1.5(a). Next, we consider the groups of automorphisms in an exact sequence. Let A be an object in A. Then we obtain by the properties in Definition 4.1.5 the exact sequence

(4.1.7) D(1AA) _ AutA(A) -k > AutB(AA) IFH(lAA). Here A is the homomorphism of groups induced by A and 1 +is the homomorphism of groups given by 1+(a) = 'AA + a. Moreover, the function ® is defined by ®(f) _ (f -') * ®A_ A(f ). In fact, ® is a derivation of groups with ®(fg) = ®(f )8 + ®(f ). Here we set xg = g*(g-') * (x) for x E H(11A)- Compare (IV.4.11) Baues [AH].

(4.1.8) Lemma A functor A in an exact sequence (Definition 4.1.5) reflects isomorphisms.

Compare (IV.4.11) in Baues [AH]. The lemma implies that a weak linear extension is a detecting functor. Recall that RealA(B) for an object B in B denotes the class of realizations of B in A; see (1.1.6).

(4.1.9) LemmaLet A be a functor in an exact sequence as in Definition 4.1.5 and assume RealA(B) is not empty. Then the group H(1B) acts transitively and effectively on RealA(B) by Definition 4.1.5(e). In particular RealA(B) is a set. 4 THE CW-TOWER OF CATEGORIES 121

Compare (IV.4.12) in Baues [AH].

4.2 Homotopy systems of order (n + 1)

Homotopy systems of order (n + 1) are triples (C, fn+1,X) consisting of a chain complex of free abelian groups C, an attaching map and an n-dimensional CW-complex X". Such homotopy systems (defined more precisely below) are motivated by the following properties of CW-complexes. Let X be a CW-complex with trivial 1-skeleton X' _ *. Then the cellular chain complex C = C * (X) is given by Cn = C"(X) = H"( X", X"-') with the boundary d: C,, Cn_ ,defined by the triple (X", X"- 1, X n " 2). Since X1 = * we have the Hurewicz isomorphism h in the composition

(4.2.1)

h 1mn+1(Xn+1,Xn)_aIrn(Xnl where d is the boundary in the homotopy exact sequence of the pair (Xn+', X"). One readily checks that fn+, satisfies fn+1d = 0. The set Zn+1 of cellular of (n + 1)-cells in X is a basis of the free abelian group Cn+1 (n + 1)-chains. Therefore fn+, describes the homotopy class of a map

f"+1: M(C,, +1,n) = v S" -)I X" (1) which is the attaching map of (n + 1)-cells in X; in fact one has a homotopy equivalence under X" -C c: Xn+1 f (2) where the right-hand side is the mapping cone of f =fn+1.By definition of fn+, and d we see that the diagram f., I Cn+1 -- 1Tn(X") dl 11 (3) C. "(X",X" commutes, where h again is the Hurewicz isomorphism and where j is from the homotopy exact sequence of the pair (X", X").). Recall that CW is the category of CW-complexes with trivial 0-skeleton and of cellular maps. Let CW" be the full subcategory consisting of n- dimensional CW-complexes. Moreover morphisms in the quotient category 122 4 THE CW-TOWER OF CATEGORIES

CW/= are 0-homotopy classes of cellular maps, where a 0-homotopy is a homotopy running through cellular maps.

(4.2.2) DefinitionLet n >_ 2. A (reduced) homotopy system of order (n + 1) is a triple (C, fn+,' X") where X" is an n-dimensional CW-complex with trivial 1-skeleton and where C is a chain complex of free abelian groups which coincide with C,, X" in degree s n. Moreover

n+P n+I 7r"Xn 1) is a homomorphism of abelian groups which satisfies the cocycle condition

fn+1d=0 (2) and for which diagram (4.2.1X3) commutes. A map between homotopy systems of order (n + 1) is a pair (f, 71),

(y,O:(C, fn+,,X")- (C',gn+1,Y") (3) with the following properties. The map 71: X" ---> Y" is a morphism in CW/= and 6: C -> C' is a chain map which coincides with C,,, 71 in degree s n and for which the following diagram commutes:

Cn+1 --a Cn+1 If.I 1a-1 (4) 1T"Xn - 7r"Yn

Let Hn+, be the category of such (reduced) homotopy systems of order (n + 1). Clearly composition is defined by Q, inX e', 7') = (W', ,,TM,). By (4.2.1) we have the obvious functors

(4.2.3) CW2/= Hn+Ia *H. with Ar"+ 1 = r". Here CW2 is the full subcategory in CW consisting of CW-complexes with trivial 1-skeleton. Clearly the functor rn+, takes X to the triple rn+IX=(C.X,fn+1,X") see (4.2.1), and the functor A carries (C, to (C, f",X"-') where X"-' is the (n - 1)-skeleton of X" and where f" is the attaching map of n-cells in X". For the definition of the homotopy relation on the category Hn+, we need the coaction µ which is a map under X"-'

(4.2.4) µ: X" -+X" V M(C",n). 4 THE CW-TOWER OF CATEGORIES 123 Here M(C,,, n) is the Moore space of C, in degreen. The coaction µ is obtained by the corresponding coaction on the mapping cone Cf with f= f", see (4.2.1)(2), that is µ = (c V 1)µf c' where c' is a homotopy inverse of c under X'- 1. The coaction µ induces an action +on the set of homotopy classes [X",Y] in Top* /=, namely (4.2.5) [X",Y] X E(X",Y) --- [X",Y] with F + a = p.*(F, a). Here we set

E(X",Y) = [M(C",n),Y] = Hom(C",,7r"Y). (1)

Since we assume that X" has trivial 1-skeleton the action (4.2.5) inducesan action

[X",Y] X H"(X", ar"Y) -- [X",Y] (2) with F + {a} = F + a. The isotropy groups of this action are considered in Section 4.8 below.

(4.2.6) DefinitionLet (6,17),(f',,'):(C,fn+,X") - (C',gn+1'Y") be maps in Hn+1. Weset Q, 17) = (i', n') if there exist homomorphisms a,+,: C1 --* Cj+1, j >_ n, such that:

(a) {71} + gn+ 1 an + 1 = {171;and (b) fk-ek=akd+dak+1,k>n+1.

The action + in (a) is defined by (4.2.5) above; {71} denotes the homotopy class of q in [X", Y'], that is in Top*/=. We call a: (t=, ii) = (e', 71') a homotopy in Hn+1 One can check that the homotopy relation is a natural equivalence relation on Hn+ 1 and that the functors in (4.2.3) induce functors

(4.2.7) CW2/= - Hn+1/= -y Hn/=.

(We refer the reader to Baues [AH] where we actually study homotopy systems in any cofibration category. Reduced homotopy systems as in Defini- tion 4.2.2 are considered in the final chapter of Baues [AH].) Let Chain2 be the category of chain complexes C of free abelian groups with C; = 0 for i 5 1. We observe that the forgetful functor

(4.2.8) C* : H3 - Chain2, (C, f3, X 2) _ C/Co 124 4 THE CW-TOWER OF CATEGORIES is actually an isomorphism of categories and that H3/= = Chaln2/= is given by the usual homotopy relation for chain maps. This way we can identify r3 with the classical cellular chain functor (reduced)

(4.2.9) (f. = r3: CW2/- -+ Chain2 = H3.

Hence the functors r"+1 and A in (4.2.7) lead to a sequence of functors which decompose the chain functor. We study the properties of this sequence in the next section.

4.3 The CW-tower of categories

The categories of homotopy systems introduced in Section 4.2 above form a sequence of categories and functors (n >- 3)

C* : CW2/= ---* H.,11= A ' H"/= - ... A H3/= = Chainz/= such that the composite is the cellular chain functor C*. We now show that each functor A is embedded in an exact sequence as discussed in Section 4.1. We call the collection of these exact sequences the CW-tower of categories. We first observe that the Postnikov functor P" which carries X to its n-type P, ,X admits a factorization

(4.3.1) P": CW2/=r- -H"+,/= P n-types.

This is clear since P" X = P" X"+' is given by the (n + 1)-skeleton of X and r"+1X determines the homotopy type of X"+' under X" by (4.2.1X2). Next we consider Whitehead's functor F. which carries a CW-complex X to the group 1'" X = image(ir" X" -' - ir" X" ). This functor admits a factorization through r". In fact, there is a functor

(4.3.2) F,,: H"/= -> Ab with I' r, X = r,, X. We define I' Y for an object Y = (C', g", Y") in H" as follows. Let Y" be a CW-complex such that g" is the attaching map of n-cells in Y". Then Y"is well defined by (g",Y"-') up to homotopy equivalence under Y"-'. Hence T"Y= 1'"Y' is well defined. A map Q, rt): X - Y in H,, admits an extension rl: X" - Y" of 31 so that also the induced map r"(f, -q) = I'"ijis well defined. It is clear thatF,,in (4.3.2) factors through the functor

P.- 1:H"/= -+ (n - 1)-types 4 THE CW-TOWER OF CATEGORIES 125 given by (4.3.1), that is rnY= rn Pn-, Y for an object Y in H. We use the functor F,, in (4.3.2) for the definition of the bimodule (4.3.3) H-F,,: (H,,/=)°P X H"/= -> Ab which carries a pair (X, Y) of objects in H,, to Ht F,,(X,Y) = Hm(X, FY). Here we set Hm(X, -) = H n(C, -) for the object X = (C, f", Xn-1) in H". We are now ready to state the following theorem which establishes the CW-tower of categories.

(43.4) TheoremThe functors A in (4.2.3) and (4.2.7) are part of the following commutative diagram in which the rows are exact sequences in the sense of Section 4.1, n >- 3.

H"r,, +-' H"+1 -L H" -®Hn+1I'"

11 lq,,,I 11 Iq" H"r,, + -->H"+1/= -A->H"/-"' H"+,rm

Here q,, is the quotient functor and 1 denotes the identity. The functor qn+ 1 is equivariant with respect to the action of H"rn. We describe the action and the obstruction operator explicitly below. (The theorem is proved in a more general form in VI.5.11 of Baues [AH], see also 11.3.3 in Baues [CH].) With the notation in Definition 4.1.5(f) the exact sequences in Theorem 4.3.4 yield towers of categories which approximate CW2/-- and CW2/= respectively. In particular we obtain the tower of homotopy categories CW2/= I

H"rn -'Hn+1/= IA H,,/= ® Hn+1rn

H3r3 ---+-4 H4/-

lA H3/= 14, H4r3 126 4 THE CW-TOWER OF CATEGORIES which somehow resembles the Postnikov tower of a space since we have obstructions and actions as we are used to in Postnikov towers. We now describe the obstruction operator. Let X,Y be objects in H"+, and let (6, r)): AX - AY be a map in H. Then an element

(4.3.5) ®x.Y(,71)EHn+I(X,I'"Y) is defined such that ®x 71) = 0 if and only if there exists a map (e, rl): X --* Y in H"+ 1 with A(6, ii) = (-rl). We define the obstruction &x, r(71) in (4.3.5) as follows. Since (f, rl)is a map in H" we can choose a map F: X" _ Y" in CW/= which extends i and for which C* F coincides with in degree < n. The diagram

C"+1

,jgn+i f-.l (1) 7rnX" F. ITnYn need not be commutative. The difference

6F(F)_ -g"+I 6n+I +F*fn+l (2) maps C"+to the subgroup f"YC -rr"Y" and this difference is a cocycle in Hom(C,,+ F,,Y). The obstruction

rl) = (®(F)} E Hn+1(X, FY) (3) is the cohomology class represented by the cocycle ®(F). This class does not depend on the choice of F in (1). Moreover, ®x'y(f, rl) depends only on the homotopy class of (6, il) in H,,/=. In addition the properties in Definition 4.1.5 (c), (d), (e) are satisfied. Next we consider the action + in Theorem 4.3.4. Let X, Y be CW- complexes with trivial 1-skeleton or objects in H",. For n < m we denote by [X,Y]" the set of all morphisms X0 - Yo in H"/= where X0 and Yo are the images of X and Y respectively in the category H". Here we use the functors in the CW-tower. The functor A yields the function

[X,Y]n+l [X,Y]". (4.3.6) A: -a Now let X and Y be objects in H"+ I. Then the action + in the bottom row of the commutative diagram of Theorem 4.3.4 is of the form

(4.3.7) [X,Y]n+1X H"(X, I'"Y)+- [X,Y]n+l and satisfies A f = Ag if and only if g =f + a for an appropriate a. We 4 THE CW-TOWER OF CATEGORIES 127

describe the action as follows. Let -q): X -> Y be a map in H"+1 and let {a} E H'(X, r,,Y) be the class represented by the cocycle

a E Hom(C", r"Y). (1)

Then we obtain by i: F Y c it"Y" the composite i s such that 71 + i a with

i+ia:X""' X"vM(C",n) Y" (2) is a map in CW/_ defined by µ in (4.2.4). We now set

{(6, r )} + {a} _ {(C,,q +ia)) (3) where ,1)} E [X, Y]"+' is the homotopy class of (6, rl) in H"+1. In Baues [AH] we check that (3) yields a well-defined action in (4.3.7). Moreover we get the action in the top row of the commutative diagram of Theorem 4.3.4 by

(6, {a} = (f, 19 + i a ). (4) Here in fact (f, 71 + ia) depends only on the cohomology class { a). The actions in (3) and (4) satisfy the properties of Definition 4.1.5(a), (b). In Section 4.8 we study the isotropy groups of the action + in (3). For CW-complexes X,Y with trivial 1-skeleton the CW-tower yields the following diagram of exact sequences of sets [X,Y]

H"(X,r"Y) [X, Y]"+'

lA (4.3.8) [X,Y]" ® H"+'(X, FY)

H3(X,r3Y) [X,Y]`

IA [X,Y]3 H4(X,r3Y)

Here [X, Y]3 is the set of homotopy classes of chain maps e* X Y. Exactness means that

(4.3.9) kernel(69) = image(A) 128 4 THE CW-TOWER OF CATEGORIES and A(f) = A(g) if and only if there is an a with g = f + a. Moreover, for an N-dimensional CW-complex X = X ' the map

rn: [X,Y] _,..+ [X,Y]n is bijective for n = N + 1 and is surjective for n = N. This follows from the definition of Hn/=. Next we derive from the CW-tower a structure theorem for the group of homotopy equivalences. For a CW-complex X in CW2 let Aut(X) _ (F(X) c [X, X] be the group of homotopy equivalences of X. Moreover, let E,,(X) C [X, X]', n >_ 3, be the group of equivalences of rnX in Hn/=. Then the CW- tower yields the following tower of groups where the arrows ® denote derivations and where all the other arrows are homomorphisms between groups.

Aut(X)

Hn(X,rnX) " E,,+1(X)

IA (4.3.10) E,,(X) s- Hn+'(X, rnx)

H3(X,r3X) -' . E4(X)

IA E3(X) f- H4(x,r3x)

Here we define the derivation ® by the obstruction ® as in (4.1.7) and we set 1 +(a) = 1 + a where 1 is the identity and where 1 + a is given by (4.3.7). We have exactness

image(1+) =kernel (A),image(A) = kernel(®).

Moreover, as in (4.3.9) we see that for X = X ^' the homomorphism

(4.3.11) rn:Aut(X)-+E,,(X) is an isomorphism for n = N + 1 and is an epimorphism for n = N. Finally we derive from Lemma 4.1.8 the following Whitehead theorem for homotopy systems. 4 THE CW-TOWER OF CATEGORIES 129

(4.3.12) Lemma A map Q,'q): X- Y in H,, is a homotopy equivalence in H"/= if and only if C* : H. X - H*Y is an isomorphism. Herewe set H" X = H,,C for X = (C, f", X -I)'

4.4 Boundary invariants for homotopy systems

We have seen that Whitehead's groups r,,Xare also defined for homotopy systems X in H". In the same way we obtain the r-groups with coefficients in A r,"_ (A, X) C I'" _,(A, X) for such homotopy systems, see Section 2.2. Here r,,_ ,is a bifunctor

(4.4.1) r,;_,: Ab°P X H,, --+Ab which fits into the binatural exact sequence

Hom(A,r;;_,X).

Recall that r,",-,is the kernel of i,,_ ,X: r"_ ,X - ir"_ ,X, see Definition 2.2.9. Here we define r,,- ,X for a homotopy system X = (C, f", X" -') by r,,_, x = r" _ , X"-1. We define the homology of X by the homology of C, that is H,,X = H"C. Let b"+,: H"+,X -> r,,X be a homomorphism and let i be the quotient map in the exact sequence b="r"X H., IX `-.ir"X-.O. Then we define C"_ (A, X) by the push-put diagram (compare (2.6.6))

(4.4.2) Ext(A,ir"X)}-°- C."_,(A,X) - Hom(A,r;;_,X)

Hence C"_,(A,X) is functorial in A and functorial in X for those maps F: X - Y in H,, for which the diagram

H"+IX CH"+IY

F.. 111 r" x r" Y commutes. We call such maps b"+, proper. Now let X = (C, f"+,, X") be a 130 4 THE CW-TOWER OF CATEGORIES homotopy system of degree (n + 1) in H,,,, and let AX = (C, f", X"-') be the corresponding homotopy system of degree n in H,, given by the functor A: H,,+ I -->H,,; see (4.2.3). We associate with X boundary invariants

bn+1 =bn+IX EHom(H,+1X,1'"(AX)), (4.4.3) 16n = )3,XE Cn_1(H,,X, AX) with µ/3nX=b"X:H,, X-->AX cFn_IAX.

The abelian group Hom(H,, + 1 X, T" AX) is determined by AX and the abelian group En _I(H" X, AX) is determined by the pair (AX, bn+ I X) as in (4.4.2). We define the secondary boundary homomorphism bn+IX by the following commutative diagram where Z, Iis the group of (n + 1)-cycles in C.

f iTrtXn C11+1 U U Zn+ I - - -1711(Ax) (1) l',.1 ,Hn+1X

Here q is the quotient map for the homology H,, + I X = H,, + IC. We observe that brt+1 is well defined by the cocycle condition of Definition (4.2.1)(3), since the kernel of j in (4.2.1X3) is I'n(AX). We can choose a CW-complex X" with n-skeleton X" and attaching maps frt+1, that is, Xn+' is the mapping cone of f, 1: M(Cn+1,n)- X", see (4.2.1). Then bn+ 1q above coincides with Whitehead's secondary boundary bn+1X"+' of Xn+'; compare (2.1.17). Using the CW-complex X"+' we define the boundary invariant )3,,X in (4.4.2) by the corresponding boundary invariant AX` + 1in (2.6.7), that is

13,X = (3,Xn+1E Cn_1(HX, AX). (2) Here the right-hand group coincides with Cn_ 1(HX"+', X"+') used in (2.6.7) for the definition of f3nX"+' By naturality of boundary invariants we get:

(4.4.4) PropositionLet F: X - Y be a map in H,, + I /= and let F: AX -I Y be the induced map in Hn/=. Then we have the equations

(a) (F,F)*b,,+1X=(Hn+1F)*bn+1Y; (b) F * IB,X = (H,F)*/3,,Y.

Here (a) shows that F is b -proper so that F* in (b) is well defined; see (4.4.2). 4 THE CW-TOWER OF CATEGORIES 131

Proof For F = Q, 71) there is a map F"+': X"+' -> Y"+' which extends r) and for which C,, F"+ 1coincides withl;in degree < n + 1. Thus the naturality of boundary invariants with respect to Fn+ Iyields the result, see Theorem 2.6.9.

We now study the realizability of boundary invariants.

(4.4.5) TheoremLet X be an object in H" with secondary boundary bnX: H" X --> 1;;_ 1X C Fn_ 1X. Then for each element

bn+1 E Hom(Hn+1X,r,,X) and for each element

'6nEC"_1(H"X,X) withµf3"=b"X

there is an object X in H"+ with A X = X and b" + 1X = b,,,and fan X = /3n. Proof Let X = (C, f X" -'). For b1 and f3" we find a map v: V --> X"-' as in the construction of the boundary operator in Addendam 2.6.5 and and /3"in the Definition 2.3.5(16). We choose v compatible with bn+1 statement of the theorem, see Definition 2.3.5(16). Then the mapping cone of vyields the CW-complex X' = C,,with bn+ 1 X"+' = bn+ i qand 'enX"+'_ /3n. Now let fn+1 be the attaching map of (n + 1)-cells in X"+ Then X = (C, fn+1, X") satisfies theproposition where X" is the n-skeleton of X"+' By definition of v the skeleton X" is also obtained by the attaching map fn in X so that AX = X.

4.5 Three formulas for the obstruction operator

We show that the boundary invariants in Section 4.4 can be used to compute the obstruction operator in the CW-tower. Let X and Y be objects in Hn+, and let F: AX -)- AY be a map in H,,. Then the obstruction element

(4.5.1) ®X.y(F)EH"+I(X,T,Ay) is defined with the property that 1(F) = 0 if and only if there is a map FO: X --> Y with AFo = F; compare (4.3.5). The cohomology group in (4.5.1) is embedded in the universal coefficient sequence

(4.5.2)

Ext(HnX,F,, AY)N H"+'(X,1'nAY) - Hom(H,+1X,r,,AY). We use the operators 0 and µ in this short exact sequence in the next 132 4 THE CW-TOWER OF CATEGORIES theorem in which two formulas describe the relation between the obstruction element (4.5.1) and the boundary invariants (4.4.3).

(4.5.3) TheoremLet X and Y be objects in H"+ Iand let F: AX --9 AY be a map in Hn/=. Then the element µ®x.y(F) is the difference of homomorphisms in the diagram H,tF H+1X H., 1Y (a) b"+tX1 r"AX - r"AY that is µ®x,y(F) = (r"F)(b"+1X) - (b"+1Y)(H"+,F).

If diagram (a) commutes then the following equation holds in Ext(H" X, i t" AY) where

H"+1Y r,, AY `- ir"AY-0 is exact:

(b) - i,k0-I ex,y(F) =0-'(F* R"X - (H"F)*p"Y).

Here the left-hand side is given by A in (4.5.2) and is well defined since we assume that diagram (a) commutes. The right-hand side is obtained by 0 in (4.4.2) and is well defined since F is b,,-proper by Proposition 4.4.4(a).

Proof Let X = (C, f"+,, X") and let Y= (C', and let 71':X"-Y" be a map associated with F = (r;, ij): AX --b AY, that is,71' extends rl and C*(-q') coincides with 6 in degree

®(F)= -g"+IS"+I+7*fn+I (1) in Hom(C"+ ir"Y"). This cocycle actually maps to the subgroup r"Y c Tr"y". Moreover µ®x y(F) is represented by the restriction

µ®x.y(F)q = ®(F) i Z"+I (2) where q: Zn+I -,,H"+,X is the quotient map. Now (a) is an easy consequence 4 THE CW-TOWER OF CATEGORIES 133

of the following diagram in which all subdiagramsexcept the one in the middle commute.

Zn+l n+1

C11+1------4c, (3)

I If I8- (b,,1Y)q

? 7TfY" nXn 17I

1'nX r,, Y Now assume that the diagram in (a) commutes. This implies that the exterior square of (3) commutes. Hence we get via the exact sequence

0 --Zn+1-Cn+1-Bn-'0 (4) the diagram

B,, _4 B,,n 419 f-11 S 14,l (5) 7TnXnIfn+1Zn+1 inYn/gn+lZ, as a quotient of diagram (3). Here and gii+1 are induced by frt+1 and gn+ 1 respectively since we use (4). The difference

A_ -gn+1CSnB+q*fn+1 (6) maps to the subgroup

ir,,AY =cokbn+lYC7Ynlgn+1Z;,+1 (7) Moreover, A represents the element

(A) =i*A-1®x Y(F) E Ext(H,,X,i1'nAY). (8) Now let X' and Y' be the mapping cones of

.fn+1 I Zn+1:M(Zn+1,n)- Xn and

gn+I I Zn+1'M(Znr+I,n - Yn respectively. Since the exterior square of diagram (3) commutes we can find an extension

1711 : X' -+ Y' (9) 134 4 THE CW-TOWER OF CATEGORIES of 17' such that Cn+ (-q") = Moreover we have isomorphisms

.7T,, X' = irrX'/fn+IZn+I and (10) 7T" Y, = ,7'X 1gn + 1 Zn + I

Using (5) and (10) we have the following diagram in which cpis a map between Moore spaces which induces H F in homology

M(H,,,n - 1) ° -M(H,,,n - 1)

M(Bn,n) £B M(B;,,n) (11)

13 Y

Yn-1

Hereq: M(H,,, n - 1) -- M(B,,, n)isthepinch map since weobtain M(H,,, n - 1) by the presentation

0 - Bn-->Zn-->H,.0 where Bn and Zn are free abelian groups. Moreover fi (resp. f3') is chosen for X (resp. Y) as in Definition 2.3.5(15). By (2.6.8) the map /3 (resp. /3') represents the boundary invariant f3X (resp. 8,,Y). All small subdiagrams of (11) except the one in the middle homotopy commute. This shows that -0, with A in (b), also represents the right-hand side of the equation in (h) and hence the proof of this formula is complete by (8).

Let X = (C,fn+1, Xn) and Y= (C',gn+1,Yn) again be objects in Hn+1 and let F = 6,,q): AX - AY be a map in H. Thus : C - C' is a chain map and -q: X ' -> Yn -' is given by a cellular map for which C. 11 coincides with f in degree < n - 1. Let

(4.5.4) (q6) E Ext(HnX, Hn+1Y) be represented by a homomorphism 3 which is part of the following composition

d s Cn+I -. Bn -LZ'+I - Cn+

Here d is the boundary in C and i is the inclusion of cycles in C'. Moreover q: I - Hn + 1Y is the quotient map so that q6 represents an element in 4 THE CW-TOWER OF CATEGORIES 135

the Ext-group (4.5.4). Using S we define the following Sdeformation+ S of the chain map f, namely let f + S be the chain map

(4.5.5)

and + S : C -+ C'with Sl (f+S ); otherwise.

One readily checks that (6+ S, -q): AX - AY is againa well defined map in H. for all S. Our third formula describes the obstruction for this map. Recall that F = (f,,q) is a bn+1-proper map if the diagram in Theorem 4.5.3(a) commutes.

(4.5.6) PropositionLet X and Y be objects in Hn+, and let (,=, r1): AX- ,1Y be a bn+1 proper mapin H. Then also + 6, r)) is a bn+1 proper mapand one has the formula

0-'®x,yQ+ S, 71)_ 71) - (bn+,Y)*(4S) in Ext(H,, X, rn AY). Here 0 is the operator in (4.5.2).

Proof Clearly (f+6,71) is bn+-proper since Hn+1(6)=Hn+i(6+S). Now the formula is an easy consequence of the fact that the following diagram commutes

n j+' gn nT 0 rnY Zn+1 --t Hn+1bn.it'

Remark In Baues [CH] 11.5.4 and (11.5.6) we have shown that the formulas of Theorem 4.5.3(a) and Proposition 4.5.6 have a generalization for the non-simply connected case. The formula of Theorem 4.5.3(b), however, is not available in the non-simply connected case. The formula in Baues [CHI 11.5.4(2) corresponds via Theorem 2.6.9(c) to the formula in Theorem 4.5.3(b).

4.6 A-Realizability Using the formula in Section 4.5 we obtain a crucial result on the A- realizability of maps in H. This leads to a classification of homotopy types of objects in Hn 1. 136 4 THE CW-TOWER OF CATEGORIES

(4.6.1) TheoremLet X and Y be objects in Hn+ and let F = (, ii): AX- AY be a map in H. Moreover assume that F satisfies the equations

(a) (FnF)*bn+iX=(Hn+,F)*bn+IY; and (b) F* f3nX = (HnF)*ppY.

Then there exists (qS) E Ext(HnX, H, Y) such that the S-deformation Fs = ( ,= + S, 71) of F in (4.5.5) is A-realizable by a map F: X --> Y with AF = FS.

In Proposition 4.4.4 we have seen that (a) and (b) are always true if F is A-realizable. Now Theorem 4.6.1 shows that these equations are the criterion for the A-realizability up to 8-deformation.

Proof of Theorem 4.6.1The exact sequence

Hn+tY rfAY (1) b,,, Y}; induces the exact sequence of Ext groups

Ext(H,,X, Hn+,Y) = rrAY) --> Ext(H,,X,iFFAY) -* 0.

(2)

Now (a) implies by Theorem 4.5.3(a) that µ®x y(F) = 0 and (b) implies by Theorem 4.5.3(b) that for i * in (2) the element

i * 0-`®x,y(F) = 0 (3) istrivial.Hence by exactness of (2)thereisan element(qS) E Ext(H,, X, Hn+,Y) with

0(bn+IY)*(q8} =®x,r(F). (4)

Now Proposition 4.5.6 shows that the S-deformation of (i;, q) satisfies

®x,r(f+S,T1) and this element is trivial by (4). Hence the obstruction property in (4.3.5) shows that there is a map F = (+ X -* Yin Hn+iwith AF = ( + S, rl). 0

We use the equations (a) and (b) in Theorem 4.6.1 for the definition of the following category. 4 THE CW-TOWER OF CATEGORIES 137 (4.6.2) DefinitionObjects in the category Hb/= are triplets where X is an object in H. and where bi+1 and /iare elements

X), (1)

with (2) The groups in (1) and (2) are defined in (4.3.2) and (4.4.2) above. A morphism F: (X, (Y, b;+1, 0;,) between such triples is a map F: X - Y in H,,/= for which the equations

(Hn+1F)*bn+1 and

F*/3R=(H,, F)*/3,, (3) are satisfied. By Proposition 4.4.4 we have the functor

(4.6.3) Ab: Hn/= which carries X in H+ 1to the triple Ab(X) = (AX, b + 1 X, /3 X) given by the boundary invariants of X in (4.4.3). This functor is not full but satisfies by Theorem 4.6.1 the realizability condition for maps up to S-deformation. Moreover by Theorem 4.4.5 this functor satisfies the realizability condition for objects. This yields the following classification of equivalence classes of objects in the category

(4.63) TheoremThe homotopy type of an object X in 1/= is completely determined by the triple (AX, b,,,, X, (3 X). In fact, the functor Ab above induces a 1-1 correspondence between equivalence classes of objects in H,,,/= and equivalence classes of objects in Hb/=.

Proof Surjectivity of the correspondence follows from Theorem 4.4.5. We now check injectivity. Let X and Y be objects in and let

F= (e,71): (AX, ,8»X) -> be an equivalence in Hb/-, that is F: AX - AY is a homotopy equivalence in H,,/= which satisfies the equations (a) and (b) in Theorem 4.6.1. Then Theorem 4.6.1 shows that there is an element 6 and a map F = (%+ S, i): X - Y with AT = (6+ S, q). Here in fact F is a homotopy equivalence in

1 since F induces an isomorphism in homology; see Lemma 4.3.12. In fact we have H*F=H*( +5,i)=H*(C+6)=H*(4)=H*(f,ii)=H*F where H* F is an isomorphism since F is a homotopy equivalence. 138 4 THE CW-TOWER OF CATEGORIES 4.7 Proof of the boundary classification theorem

In this section we complete the proof of the classification theorem 3.4.4. It remains to prove that A' in Proposition 3.7.5 is a detecting functor. We again assume that m >: 2 and that

(4.7.1) C = typeset with n = m + r. The bype functor F on C is given by F(H, X) X). We consider the functor

A': spacesm ' -, Bypes(C, F)with (4.7.2) A'(X) _ (P"-,X, H,, X, Hn+I X, b"+I X, #rtX )

(4.7.3) TheoremThe functor A' is a detecting functor.

This is the reformulation of Proposition 3.7.5. It is enough to consider the case m = 2. Then we get the equivalence of categories )"+ ' /_ (CW2 -+ spaces2+ Here (CW2)"+' is the category of CW-complexes X with X' and dim X:5 n + 1 and of cellular maps. We obtain a proof of Theorem 4.7.3 by use of the following commutative diagram of functors.

(CW2)"+ I/_ n'-0Bypes(C, F) (4.7.4) JA l b (Hrt+1/=)n+1 x (Hn/=)

Here (Hrt+,/=)"+' denotes the full subcategory of Hrt+I/= consisting of objects X = (C, with dim(X) 5 n + 1 or equivalently with C, = 0 for i > n + 1. Similarly the category (H,'/=)"+' consists of (n + 1)-dimensional objects in Hb. The functor Ab in the bottom row of (4.7.4) is a restriction of the corresponding functor Ab in (4.6.3). Moreover the functor r in (4.7.4) is the restriction of the functor rrt+, in (4.2.7). The CW-tower shows that r is a detecting functor. Finally we obtain the functor A" in (4.7.4) as follows. For this we use the functor

(4.7.5) P"-,: H"/= -+ (n - 1)-types defined as in (4.3.1). Then A" carries an object (X, b"+,, in Hb/= to the object (4.7.6) A"(X,brt+,")=(P"-IX,H"X,,B",H"+1X,brt+d. 4 THE CW-TOWER OF CATEGORIES 139

The definition of A' in the classification theorem and the definition of Ab in (4.6.3) show that diagram (4.7.4) commutes. For this we use identifications

f,XF,Pii-1Xfori

rn-1(A,X)rn-1(A,PiiX) (2)

which are available for objects X in H,, in the same way as for spaces X. Now it is obvious by (4.7.5) and (4.7.6) how to define A" on maps, namely

A" (F) = (P, -, F, H,, F, H,,,, F). (3)

Using the naturality of the identifications (1) and (2) we see that A" as a well-defined functor. Below we show

(4.7.7) TheoremThe functor A" is a detecting functor.

Using the proposition and the results in Section 4.6 we can show that also A' in (4.7.4) is a detecting functor as follows.

Proof of Theorem 4.7.3Let X0 be an object in Bypes(C, F). We first find a A"-realization X1 by Theorem 4.7.7, then we find a A"-realization X, of X1 by Theorem 4.4.5, and then we get an r-realization X3 of X2 since r is a detecting functor. Hence X3 is a A'-realization of X0 by the commutativity of (4.7.4). The Whitehead theorem shows that A' reflects isomorphisms hence it remains to show that A' is a full functor. For this let X3, Y3 be objects in (CW2)n+ 1/= and let X, = r(X3), X1 = A'(X,), X0 = A"(X,) = A'(X3), and let Y2, Y1, and Yo be given accordingly by Y1. Then any map FO: X0 -* Yo admits a A"-realization F, = (6, rl): X1 -> Y. By Theorem 4.6.1 there exists S such that F,' = (C+ S, q) admits a Ab-realization F,: X2 - Y, which in turn admits an r-realization F3: X3 -> Y3. Since e + 6 induces the same homology homomorphism as 6 we see that

A"(C+3,i)=A"(6,17).

Hence we get A'F3 = F0 and thus A' is full. 0

Proof of Theorem 4.7 We first consider the A"-realizability of objects. For this let Y be a 1-connected (n - 1)-type, and let H", H"+, be abelian groups with H"+, free abelian. Moreover let b,, be a homomorphism for which the sequence H"Fn-IYY "Y (1) 140 4 THE CW-TOWER OF CATEGORIES isexact; compare Definition3.2.2(1). We construct an object X = (C, f", X"-') in (H"/=)"+' which realizes the tuple (Y, H", H,1, b") that is, there are isomorphisms

H"+1= Hn+1C (2) Hn = H"C (3) Y= P"+1X (4) b"= b,, X:H"X-- Fn-1X=I'rs-1Y. (5)

We construct X as follows. Let Z,, be a free abelian group and let q: Z" -* Hn be a surjection with kernel B. Then we define by the cellular chain complex C,,Y of Y with cycles Z,,Y and boundaries BY the chain complex C as follows:

Cn+1 = H"+1 ®B" (6) Cn= Z"®B"-1Y (7) Cf=CJY for j

The boundary d: C,,+, -- C" is trivial on H,, 1 and is the inclusion B,, C Z" if restricted to B,,; moreover d: C" - C,,- Iis trivial on Z,, and is the inclusion on B,, -,Y. Then clearly (2) and (3) are satisfied. Next we define X"-' by the (n - 1)-skeleton of Y, that is X" - ' = Y"' For the construction of

f. : -1 (9) we make the following choices. Let d y be the boundary in C. Y. We choose a splitting t,

C"YOdy B"-IY (10) t with dt = 1. Moreover we choose a homomorphism b' for which the following diagram commutes

Bn C Z" -i Hbb= 1"-,Y b'I 1 B"(Y) C Z"(Y)

Here b' exists since b,,Y is injective and since the image of b"Y is the kernel of i"-,Y in (1) which in turn is the image of bn by exactness in (1). Hence b in the diagram is well defined and surjective. Therefore the lift b' exists since Z" is free abelian. In fact we can choose Z,, in such a way that b' is surjective. For this choose a surjection Z,, -A where A is the pull back of q' 4 THE CW-TOWER OF CATEGORIES 141 and b, where q' is the quotient map in (11). Using b' and t we get the surjective homomorphism

qp: C,, = Z. ® Bn 1(Y) -. Cn(Y) (12) for which cp I Zn is the composite Z b Z"(Y)CC"(Y)

for which rp I Bn_1(Y) = t. Now let (C,, Y, g,, Y"-') be the homotopy system of order n given by Y with gn:C"(Y)- 1Tn-IY' We define fn in (9) by the composite

Jn (13) Then X = (C, fn,Yrt-1) is a well-defined homotopy system of order n. In fact, fn satisfies the cocycle condition since gn does and since b'(Bn) C Bn(Y). Moreover h a = jfn in (4.2.1X3) is satisfied since we know that hd y = jgn is satisfied. We now observe that irn_ 1X = in_ IYsince tp above is surjective. Hence we get the homotopy equivalence (4). Moreover we get (5) by the definition of bnX in (4.4.3X1) and by the commutative diagram (11). This completes the proof that X = (C, fn, X"-') is a realization of (Y, Hn, Hn+1,bn). This, however,also implies that (X, bn+ 1, 13")isaA"-realizationof the object (V. Hn, Yn, Hn+ 1, bn +). For this we use the natural identifications in (4.700, (2). By the homological Whitehead theorem it is clear that A" reflects isomorphisms. Therefore it remains to show that A" is full. For this let X b = (X, bn+1, h'n)and Yb = (Y, b;,+ , /3) be objects in (Hb/'')"+' with X = (C, fn, X"-') and Y= (C', gn,Y"-') and let (F; fin, (pn+ 1): Xb --> Yb be a map in Bypes(C, F), that is

fin' H. -Hn, n+l' Hn+l Hn+1 (14) are homomorphisms (with H, = H,, X and H* = H. Y) and (15) is a map which we can assume to be cellular. Recall that the n-skeleton of Pn+ 1 X coincides with X" where X" is the mapping cone of fn; in the same way the n-skeleton of P,,_ 1Y coincides with Y" where Y" is the mapping cone ofgn. Thus thecellular map F in(15) yieldsarestriction (F", F" - '): (X", X" -') --> (Y", Y" - '). We now show that there exists a A'- realization

(6,Fn-'): X-> Y (16) 142 4 THE CW-TOWER OF CATEGORIES of (F, p", For this we choose splittings of Ck -. Bk_ land Ck -. Bk_,. Hence we get

Ck=Zk®Bk-1, Ck=Zk®Bk-1 (17) with Zn+ I = Hn+1, Z;,+, = H;,+ 1since C and C' are (n + 1)-dimensional. We consider the following diagram t q Cn Z" -* c n 1 H" -b "F. - ix 7T 1 X if.(*) lip" IF. C;, Z', -' H;,brn -1 Y C

Here C= Cn(F") is induced by F". This implies that the exterior square of the diagram commutes. Therefore & admits the restriction ,: Zn -+ Z;,, also denoted by f,. Since (F,cpn,(pn+1) is a map between bypes we see that for bn = µf3n and b;, = µ/3;, the equation

F* bn = b;,'pn (18) is satisfied. In fact, (18) is a consequence of the equation F*(8(, compare Definition 3.2.2(3). Hence all subdiagrams of the diagram above commute except possibly diagram (* ). We have however F* b" q = b' q' 6 so that the difference

->kernel b,,cH,, (19) maps to the kernel of b,,. Since b;,q' = bnY" we see that we have surjections

q0 q q'q" : ar,,Y" kernel bnY" kernel b;, and hence we can choose a homomorphism

/3: Zn -* n,Y"withq'q" /3 = 0. (20)

Let p: Cn --> Z,, be the projection. Then we obtain by the action in (4.2.5) a cellular map F" + /3p: X" -> Y" which extends F"-'. We can therefore replace f in the diagram by

Sn =C"(F" + pp) (21) 4 THE CW-TOWER OF CATEGORIES 143

with the effect that then all subdiagrams of the diagram are commutative. In particular we get for subdiagram (*) q-4'C"(F"+13p)IZ" = =gyp"q-q'1;,,-4'q"(3=0-Q=0.

Hence the restriction &B: B" -* B.', of"is well defined and we can set

SntB 4111 = 9n+ 1 ® (22) by use of (17). Moreovercoincides with C* F` in degree < n - 1 so that a chain map ,=: C --> C' is well defined. Moreover it is clear that (6, P- 1) is a A"-realizationof (F, rp", cp"+1) since we have the natural isomorphisms (4.7.6)(1), (2).

The proof of Theorem 4.7.7 above completes the proof of the classification theorem 3.4.4; compare Theorem 4.7.3.

4.8 The computation of isotropy groups in the CW-tower We consider two different actions of abelian groups on certain sets of homotopy classes and we describe a general method for the computation of isotropy groups of these actions. For an n-dimensional simply connected CW-complex X" and a space Y we have the action (4.8.1) [Xn,Y] x Hn(X", ir"Y) ± [X",Y] defined in (4.2.5)(2). Here [Xn,Y] is the set of homotopy classes in Top*/=. The cofibre sequence for j: Xn-1 cX" shows that j*: [Xn,Y] _ [Xn-1,Y] satisfies for elements 71,7)' E [ X", y ] j*(71) = j*(r)') - 3a E H"(X", 7r"Y)with + a = ii'. (1) Thus the action (4.8.1) is useful for the inductive computation of the sets [X",Y] with n z 1. Let 1(71) cH"(X 7rnY) be the isotropy group of the action in 71, that is

1(71) _ (a E H"(X", 7rnY); 71 + a = 71). (2) Hence by (1) the orbit of 71 is given by 71E (j*)-1(710) -H"(X",7rY)/1(71) (3) where 710 = j*71 and where the right-hand side is the quotient group. The bijection carries (a} to 71 + a. The group I(q) depends actually only on 710. For the computation of 1(71) in (2) one needs a spectral sequence; compare Baues [AH] VI.5.9 (4). 144 4 THE CW-TOWER OF CATEGORIES

On the other hand, let now X and Y be homotopy systems of degree n + 1 and let [X, Yin + 1 be the set of homotopy classes of maps X -' Yin Hn+ /=. Then we have by (4.3.7) the action

[X,Y]n+I xH"(X,1'Y) + [X,Y]"+l (4.8.2) The functor A: H"+ I/= -* H"/= yields the function A: [X, Y]"+' - [X,Y]" which satisfies for elements F, F' E [X,Y]"+' A(F)=A(F')p3aEH"(X,F,,Y) with F+a=F'. (1) Thus (4.8.2) is useful for the inductive computation of the sets [X, Y]", n >_ 1; compare for this the tower of homotopy sets in (4.3.8). Let I(6, 77) c H"(X, F"Y) be the isotropy group of the action in ((6,71))=F where 6: C -> C' is a chain map and where 77: X" -+ Y" is a cellular map with X=(C, f"+,,X") and Y=(C',gn+1,Y"). We have 10,77)_{aEH"(X,fY);F+a=F}. (2) Hence the orbit of F = {(f,17)} is given by FE A-'(Fo) =H"(X,rnY)/I(1;,-0) (3) where F0 = AF. The bijection carries the coset {a) to F + a. The group I( f, 77) depends only on F0. For the computation of IQ,,q) we have the following result. Let j: H"(X,FRY) cH"(X",FnY) -->Hn(X",7rnY") (4) be induced by the inclusions C* X" c C and I'nYC 7r, Y" respectively. More- over let b_jY Hn+IY F,,Y--- iF"Y->0 (5) be an exact sequence defined by the secondary boundary homomorphism bn+ 1 of Y and let 0 -> i FnY > 7rnY -> ker b,,Y --> 0 (6) be the short exact sequence given by Whitehead's exact sequence. Compare Theorem 2.6.9(c) where we show that the extension element (7r,,Y) given by (6) satisfies ( /3nY)t = OT,,Y) E Ext(ker(bnY),iFY) (7) so that the extension (6) is determined by the boundary invariant F'nY. Let T: Hom(Hn_ 1X,ker(b,,Y)) - Ext(Hn_1X,iF,,Y) (8) be the boundary homomorphism associated to the extension (6), that is T(a) = a"{7rnY} (9) 4 THE CW-TOWER OF CATEGORIES 145 for a: Hn_,X -- ker(b,,Y). As usual the sum of subgroups U U2 in an abelian group U is given by U, + U2 = (x + y; x c- U,, y E U2).

(4.83) TheoremThe isotropy group A6,77) is the following sum of three subgroups in H"(X, I''Y):

Here I(-q) c H"(X",1rnY") is the isotropy group of 19 E [X", Y"] in (4.8.1X2) and j is the homomorphism in (4) above. The homomorphism i* and A,

Ext(Hn_,X,ir,,Y):"Ext(H,,-,X,r,,Y)H"(X,r y) are given by the surjection i in (5) and by the universal coefficient theorem respectively. This shows that for the boundary rin (9) the subgroup i(i*' imageT) is well defined. Moreover (bn+,Y)*H"(X,HH_,Y) is the image of thecoefficient homomorphism (bn+,Y)*: H"(X, Hn+,Y) H"(X,rnY) induced by bn+1Y in (5) above.

Proof of Theorem 4.8.3The result in Baues [AH] VI.5.16 shows that 1(6,71) =j-'(I('q)+An(X,Y))

where A71) C image j so that I(, r1) =j- 11(77) + j-'An(X,Y). We now show

j-'An(X,Y) = (bn+,Y)*H"(X, H, Y) + O(i *' image -r). (1) This implies the proposition of the theorem. For the proof of (1) we first recallthedefinitionof thesubgroupA,,(X, Y) c H"(X", irnY").Let gn+1: Cn+1 - irnY" be the attaching map in Y. ThenAn(X,Y) is the set of all cohomology classes (gn+1 a,, +1) for all which there exist a,+1:Cj->Cj+1, j?n with (2) akd+d'ak+1 =0 fork>_n+1.

Compare Baues [AH] VI.5.16(7). Consider the diagram

d d Cn+1 _a Cn+1 Cn

(3) Cn+3dCn+2dCn+1 g irnY" 146 4 THE CW-TOWER OF CATEGORIES

Let t: B" --> C"+t be a splitting of d with B" =dC"+,. A sequence of maps (aj, j > n) satisfying (2) exists if and only if there exists a commutative diagram d tBn N C.

(4) C"+2---a Cn+l d' or equivalently if and only if a"+,(B,,)CB,,+I with B',+, =d'Cn+2. Clearly this condition is necessary by (2) where we set k = n + 1. On the other hand, this condition is sufficient since Cn+, = Z"® tB,, and since we can define aj, j>-n +2, by art + 2 1 tBn = - a

an+2 1 Z"+I = 0 (5) a,=0forj>n+3. Hence (4) shows

A,,(X,Y) = {{gn+lan+I};an+I EHom(C",C;,+,)andan+IB" cB;,+,). (6)

The attaching map g"+Iis embedded in the following commutative diagram

Z", r"Y ' »it"Y

Sn+, n qy Cn+, w y TrnY

Id' Id' (7) I

B,, >`ea ker(b,Y") -p» ker(b,,Y) n n

Z;, --P. H"Y

The vertical arrows are the obvious inclusions and surjections respectively. The row i8, p is short exact and hence a free resolution of the group ker(b,,Y). The map qy is induced by the inclusion Y" cY"+I where Y"+' is the mapping cone of gn+, and ir,Y= ir,Y"+'. All rows of the diagram are 4 THE CW-TOWER OF CATEGORIES 147

exact. Moreover all columns contain short exact sequences in the upper part. We now choose splittings t, and to, d" it"Y^ ker(bnY) rl

Cn' -d-, +1 " B,' to0 of the surjective homomorphisms d" and d' respectively in diagram (7) above. We define -8 =ir'(tlie-gn+l to)EHom(B,',,rnY) (8) Hence the following diagram commutes. irfy

I 9rr, I ker(bn Y")---- 1rn Y (9)

ker(b,,Y)= ker(b,,Y) This is clear since gygn+I= 0. By (9) we see that i/i represents the extension {i"Y} E Ext(ker(bnY), irnY). We now consider j -'An(X, Y) where j: H"(X, rnY) N H"(X", Ir,Y") (10) is actually an inclusion defined by C. X" cC and by ir: rnYc 1rnY"; see (4.8.2X4). A mapan+ 1as in (6) induces a map an +, for which the following diagram commutes

Cn /Bn ~- Cn la"., lg".,Q.., g-- (11) Cn +I /Bn +1 77n Y" Ito aIt, Hn+1(Y) ®B;, M> rn(Y) ED ker(b,Y") Here p is the quotient map and gn+1 is induced by gn+,. The isomorphisms to, t1 are induced by the corresponding splittings above. Moreover (7) and (8) show that the matrix M in (11) is given by

b"0,1Y M= I iJ. (12) a 148 4 THE CW-TOWER OF CATEGORIES

The isomorphism t, in (11) yields the following diagram with j as in (10) and where we set K = ker(b"Y").

H"(X", 1T"Y")

H"(X, K) .-H"(X,I'"Y®K)K H"(X,I'"Y) (13)

TM T(b_ ,Y,-s). H"(X,H"_,Y®B;,) K ker(O,iB)*

The definition of M in (11) together with (6) show

A"(X,Y) = image(jM*). (14)

Hence we get by a diagram chase in (13) the formula

j-'A"(X,Y) _ (b"+lY, ker(O,iB)1 (15)

Since 'B is injective one gets

ker(0,'B) * = H"(X, H"1Y) ®0 ker(i,,) (16) where i 4, = Ext(H"- ,X, i B). Moreover (9) and the six-term exact sequence for the boundary T in (4.8.2)(8) yield the following commutative diagram

Hom(H"_,X,kerb"Y)=Hom(H"_,X,kerb"Y)

Ext(H"- ,X, B;,) Ext(H"_,X,il'"Y) (17)

Ext(H"_,X,K) Ext(H"_,X,f,,Y)

The left column is exact. A diagram chase in (17) shows

f3*kerit =i-V(imageT) (18)

Thus (15), (16) and (18) yield formula (1) and the proof of Theorem 4.8.3 is complete. 5 SPANIER-WHITEHEAD DUALITY AND THE STABLE CW-TOWER

We consider some aspects of combinatorial homotopy theory which arise in the stable range. In particular, we use Spanier-Whitehead duality which carries homotopy groups to cohomotopy groups. Whitehead's certain exact sequence for homotopy groups corresponds in this way to a dual sequence for cohomotopy groups. The secondary boundary in the dual sequence is related to cohomology operations which appear in the Atiyah-Hirzebruch spectral sequence. Moreover we describe the Spanier-Whitehead dual of the stable CW-tower.

5.1 Cohomotopy groups

We introduce cohomotopy groups and we describe the dual of Whitehead's certain exact sequence which embeds cohomotopy groups and cohomology groups in a long exact sequence. To stress the duality we develop the theory along parallel lines which are dual to each other. Let X be a CW-complex with base point. Homotopy groups and cohomotopy groups are defined by the set of homotopy classes in Top*/- it"X=[S",X], 7r"X= [X,S"].

Here 7r"X in general is not a group, but in the stable range, dim X< 2n - 1, we have the suspension isomorphism 1: [X, S" ] = [ EX, S"" 1 ] which yields an abelian group structure for 7r"X. The dual of the skeleton X" is the coskeleton X" which is the quotient space

(5.1.2) X" =X/Xn-1

Hence the coskeleton of the m-skeleton, m >- n, is X,, = X n' /X n -1 which is also the m-skeleton of X" with X" cX". We point out that X is a one-point union of n-spheres or equivalently a Moore space of a free abelian group

(5.1.3) X = V S" = M(C"X, n). Z" 150 5 SPANIER-WHITEHEAD DUALITY

Here Z,, is the set of n-cells in X and C" X is given by the cellular chain complex C,, X of X. Let C*X be the cellular cochain complex of X, that is

CnX = H"(Xn, Xn-')= H"(X,") (5.1.4) C"X = Hom(C"X,71).

The coboundary d: C"X -> C"+1X is induced by the boundary d:C"+ 1 X -- C" X. We have the attaching map of n-cells

(5.1.5) f": V S"-' =M(C"X,n - 1) ->Xn-t zn

Dually we have the coattaching map

f": Xn+1 -' V Sn+1 =M(CnX,n + 1). zn

This map is obtained by the cofibre sequence

X,n -Xn -Xn+,f Ix - ... given by theinclusion X c X, of then-skeleton. We clearly have 7rn- IM(C,, X, n - 1) = CnX and 7rn+'M(C"X, n + 1) = C" X. Hence by applying the functors 7rn-and 7r"+' to fn and fn respectively we get the induced homomorphisms

(5.1.6) f":C,,X-'7Tn-1X"-1,resp. f":C"X-Tn+1Xn+1 which actually determine the homotopy classes of the corresponding maps in (5.1.5). Therefore we denote the homomorphisms (5.1.6) and the corresponding maps in (5.1.5) by the same symbol. We write

X = XN (X is an AN - N-polyhedron) if X isan M-dimensional CW-complex withtrivial (N - 1)-skeleton X" -' = *. If M < 2N - 1 we obtain the following two exact sequences of abelian groups which are `dual' to each other.

(5.1.7) 0--+rMX-' 7fMX-'HMX-Fm-IX_' ... -HN+IX-'0-+lrNX=HNX,

0-+FIX_ 7TNX--+HNX_IF N+IX _._,... -'HM+'X-0,7rMX=HMX. i h b The top row is the exact sequence of J.H.C. Whitehead. The bottom row is 5 SPANIER-WHITEHEAD DUALITY 151 defined as follows: Let M < n < N. The inclusion X n -' --> X" and the projection X. -* X"+ Iyield the F- groups F, X = Im(ir"X"-' - IT,, X'), (5.1.8) F"X = Im(Tr"X"+t - ITnX,,). Now the inclusion X" -> X and the projection X - X" induce the maps

i = !" : I'" X C 7r,,X" - 7r,, X andi - i" : f"X C ex,, -- 7r "X (1) respectively. Moreover, we have the Hurewicz homorphism

h = h" : 7r"X -> H" X= Hn(X, Z), (2) h =h":7r"X -H"X =H"(X,71) by hn(a) = a*(e"}, h"(/3) = f3*{e"}. Here {en} E H"( S", 71)and {e"} E H"(S",71) are generators which are dual to each other. Next we obtain the secondary boundaries b=bn:H"X--+rn-IX, (3) b = b": H"X _ F,1 IX by the maps in (5.1.6) as follows. Let Z"X and Z"X be the group of n-cycles and n-cocyles respectively. Then we have commutative diagrams

Z,,XCC,,X 7rrs_1Xn-'

H"X ---'=- I'n-1X,

Z"X C C"X '7Tn+1Xn+I

H"X --- n'----r"+1X.

(5.1.9) PropositionFor X = XN and M:5 2 N - 1 the sequences in (5.1.7) are exact.

Proof The sequences of J.H.C. Whitehead is extracted from the homotopy exact couple of X. In the same way we extract the bottom row of (5.1.7) from the cohomotopy exact couple which is available in the stable range. 152 5 SPANIER-WHITEHEAD DUALITY 5.2 Spanier-Whitehead duality

We recall some facts about Spanier-Whitehead duality which carries homo- topygroupsinthestablerangeto cohomotopy groups.Moreover Spanier-Whitehead duality carries Whitehead's exact sequence to the dual sequence for cohomotopy groups described in Section 5.1. Let AN-N be the full homotopy category of all finite AN -N -polyhedra or equivalently of all finite CW-complexes X = XN with dim X:5 M and trivial (N - 1)-skeleton XN- '= *. This is a full subcategory of Top*/=. In the stable range M < 2 N - 1 Spanier-Whitehead duality is a contravariant isomorphism of additive categories (5.2.1) D: AN N AN-N This isomorphism carries X to DX = X* and carries the homotopy class of f : X -' Y to the homotopy class Df = f * : Y* --* X*. The isomorphism D satisfies DD = identity, that is X** =X f** = f The functor D is determined by (N + M)-duality maps as follows. Recall that for pointed CW-complexes X, Y we have the smash product X A Y = X X Y/X V Y which satisfies S" A S' = S"+m and S' A X = T.X. (5.2.2) DefinitionLet X = XN, M <2 N - 1. An (N + M)-duality map is a CW-complex X* = (X*)'in A- N together with a map N N Dx: X* AX- SN+M (1) such that the following compositions are isomorphisms for N< q < M:

7'N+q(X*) AX, [SN+gAX,X*AX]

a i(Dx). (2) 2N.4 ,jrM q(X) = [SN+q AX,SN+M] X*A_ IrN+q(X) [X* ASN+q,X* AX]

(3)

4 FN+Q I(Dx). ,TM-q(X*) [X* ASN+q,SN+M] Spanier and Whitehead have shown that for each X in AN- N there exists an (N + M)-duality map; compare Spanier and Whitehead [DH] and Chapter 8, Exercises F in Spanier [AT]. Remark Geometrically one obtains an (N + M)-duality map D. in Defini- tion 5.2.2 as follows. Let X be embedded in S"+', n = 2M, and let Y be a 5 SPANIER-WHITEHEAD DUALITY 153 finite CW-complex which is a strong deformation retract of the complement S"+' - X with YcS"+'X. Then pick a point a c= S"+' - X U Y and consider the inclusion

XUYcS"+'-(aI= R^+1 (1) Since X n Y= 0 is empty we have

X x Y c R xR +i-0 (2) where A = ((x, x) I x E R"+'} is the diagonal of OR"+'. We obtain a deformation retraction r with ri = 1, S"-' -,,aR"+lxR"+1 r,S" (3) given by i(x) = (x, 0) for x E Sn = (x E 6R"+ 1Ilxll = 1) and r(x, y) = (x - y)/Ilx -yll The composition of r and (2) yields the map

XxY- S" (4) which is null-homotopic on X V Y. Hence this map induces a map

D:XAY=C1-S" (5) where C1 is the mapping cone of the inclusion j: X V Y c X x Y. The map D is then an n-duality map in the sense of Definition 5.2.2. There is a CW-complex X* in AN-N together with a homotopy equivalence IM NX* =Y. (6) Moreover there is a map DX:XAX* SM+N (7) for which the (M - N)-fold suspension I M_NDXis homotopic to yM N(XAX*) XA Y-, S" =S2M. The map DX is the duality map in Definition 5.2.2; compare Cohen [SH]. We are now ready for the definition of the duality functor D in (5.2.1). (5.23) Definition of D Choose for X =A" N an (N + M)-duality map and set DX = X * . For X* we can choose DX* =X and for a sphere S N+ q we choose DSN+q = (SN+q)* = SM-q. Moreover, D is defined on maps by the following commutativediagram [X,Y]- D a [Y*,X*] Y- V \n X

[Y* AX,Y* AY] 7 [Y* AX,X* AX] (DY).\ /Dx)[Y* AX,SM+N] 154 5 SPANIER-WHITEHEAD DUALITY in which the compositions (DX)*(AX) and (D y)*(Y* A) are isomorphisms.

The inductive construction of X* in Chapter 8, Exercises F, in Spanier [AT], shows that the functor D can be chosen such that the following properties are satisfied. In the following let 0 < q < M - N. For X = XN the cells eN+q form a basis of CN+qX. Let eN+q be given by the dual basis in CN+qX = Hom(CN+q,7L). Now each cell eN+q in X is in 1-1 correspondence to a cell eM_q in X*. This correspondence yields the identification

CN+qX=CM-qX*,eN+q'-" (5.2.4) eM-q. We call eM-g the dual cell of eN+q. Moreover, for skeleta and coskeleta the following equation holds:

(5.2.5) (XN+q)* = (X* )M-q; compare the notion in (5.1.2). The attaching and coattaching maps yield the commutative diagram

M(CN+q,N+q-1)* rf- (XN+q-1)*

(5.2.6) M(C,y_gX*,M-q+ 1)4 - (X*)M-q+l fM-q

Here M(A, k) denotes the Moore space of A in degree k; we use the fact that for a finitely generated free abelian group A we have M(A, N + q)* = M(A*, M - q), A* = Hom(A,7L). By Definition 5.2.3 and (5.2.4) the dual of a map a: SN+q, SN+q' between spheres is a* _ I ka. SM-q' -> S'"-q with k = M - N - q' - q. Hence by (5.2.6) the dual of the mapping cone C.= SN+qU.eN+q+ 1is the mapping cone of a* = Vu or equivalently

(5.2.7) (Ca)* =L.kCa.r If a: SN+q _SN+q isa map of degree n we obtain the Moore space Ca = M(7L/n, N + q) of the finite cyclic group 7L/n. Hence the dual of this Moore space is, by (5.2.7), again a Moore space (5.2.8) M(7L/n, N + q)* = M(7L/n, M - q - 1). From (5.2.4) and (5.2.6) we derive the commutative diagram CN+qX d CN+q+1X

(5.2.9) II II CM-qX* d CM_q+IX* 5 SPANIER-WHITEHEAD DUALITY 155 Here d is the boundary in the cellular chain complexes of X and X* respectively. This yields for any coefficient group the isomorphism

(5.2.10) HM_q(X*,G) =HN+9(X,G) which is natural in X and G. For example if X = M(A, N + q) is the Moore space of a finitely generated abelian group A then we have H8+9(X) = Hom(A,71) and HN+q+'(X) = Ext(A,7L) so that the homology groups of the dual X* = M(A, N + q)* are by (5.2.10) Hom(A,71), i=M-q H;M(A,N+q)*= Ext(A,7L), i=M-q-1 0 otherwise

This implies that we have a homotopy equivalence

(5.2.11) M(A,N+q)* = M(Hom(A, Z), M - q) vM(Ext(A,7L),M-q- 1) where the right-hand side is a one-point union of Moore spaces.

For homotopy and cohomotopy groups we have by Definition 5.2.3 the isomorphism D:

(5.2.12) ?rM_v(X*) = 7rN+q(X) which we use an identification. Similarly we get for the F-groups the isomorphism

(5.2.13) FM_Q(X*) = rN+9(X).

Now the exact sequences in (5.1.7) have the following property.

(5.2.14) PropositionFor X = XN in AN -N with M < 2 N - 1 there is the natural commutative diagram of exact sequences: 0->FMX* - irMX* -+HMX* .... -,HN+1X* - 0 -7rNX* =HNX*

II II II II II 0 -r NX --a 7rNX - HNX - ... -+ HM -' X -+ 0 ---it NX = H NX

This shows that each result on the operators in the exact sequence of J.H.C. Whitehead yields a dual result on the dual operators. Similarly, it is well known that homology and cohomology operations behave well with 156 5 SPANIER-WHITEHEAD DUALITY respect to the isomorphism in (5.2.10); see 27.23 in Gray [GT] and Maunder [CO]. For example the Steenrod squares have the property that for X = XN (M < 2N - 1) the following diagram commutes

Sq' HN+q(X,l/2) HN+q+i(X,71/2)

(5.2.15) HM-q(X*,7L/2) - HM-q_i(X*,7L/2) sq;

Since H*(X,l/2) and H,, (X,71/2) are dual vector spaces over Z/2 we can consider the 1/2-dual operation (Sqi)* = Hom(Sgi,7L/2). This operation does not coincide with Sq'. The connection is defined by the automorphism X of the Steenrod algebra: X(Sgi) = (Sqi)*; see for example Gray [HT]; however, (Sq2)* = Sq2. We denote by Sqg, Sqg the integral operations which are the composites SqH"+i(X,Z/2), Sqa: H"(X,71) H"(X,71/2) (5.2.16) SqH"(X,71) P. H"(X,71/2) - H"-i(X,71/2) where p: 71 - 7/2 is the quotient map. The Adem relation Sg2Sg2 = 0 yields the secondary operations 4)* and 4*; see Mosher and Tangora [CO], H"(X,7D D kerSg2- HN+3(X,71/2)/Sg2Hn+'(X,71/2), (5.2.17) H"(X,7L):) kerSg2-b H"_3(X,71/2)/Sg2H"_1(X,7/2).

Again these operations satisfy Spanier-Whitehead duality, that is for X = XN, M < 2 N - 1, we have the commutative diagram

HN+q(X,7)DkerSq' -> HN+q+3(X,71/2)/Sg2HN+q+'(X,Z/2)

HM-q(X*,Z)DkerSg2 HM-q-3(X*,7L/2)/Sg2HM-q-t(X*,7L/2) compare Maunder [CO].

5.3 Cohomology operations and homotopy groups There are some old results which relate the operators in the stable F-sequenceofJ.H.C.Whitehead with homology operations.Using Spanier-Whitehead duality one has the dual results for cohomotopy groups and cohomology operations. 5 SPANIER-WHITEHEAD DUALITY 157

Let Y be an (n - 1)-connected CW-space. Then 7r"Y, it"+lY, 7r"+ 2Y, ... are called the first, second, third,... non-vanishing groups of Y respectively. Dually we have for an n-dimensional CW-space X the first, second, third.... non-vanishing cohomotopy groups given by ?r "X, 7r"-1 X, Tr"- 2X,... respectively. It is a classical problem to compute for small values of k = 1, 2,3,... the kth non-vanishing homotopy groups and cohomotopy groups in terms of homology and cohomology. We here consider this problem in the stable range. Since we apply Spanier-Whitehead duality we assume that all CW- complexes considered are finite. Recall that we write Y = YN if Y is a CW-complex with trivial (N - 1)-skeleton and that we write X = X M is X is a CW-complex of dimension < M. The first non-vanishing homotopy group was computed by Hurewicz and the first non-vanishing cohomotopy group was considered by Hopf. Their results yield isomorphisms arNY=HN(Y,7L) forY=YN,N>2, (5.3.1) arMX=HM(X,Z)for X=XM,M>:I. These isomorphisms are also consequences of the exact sequences in (5.1.7). For the second non-vanishing group we have by (5.1.7) the following exact sequences (Y = YN, X = X')

bN+2 i HN+2Y FN+ 1Y ITN+IY- HN+ 1y, (5.3.2) HM-2X bm-rM-1X->7TM-1X-.HM11-1X.

(Here it is enough to consider Y = YN + 2 and X = XM_ 2,M = N + 2, so that X and Y are AN-polyhedra.) In the stable range there is the isomorphism

(5.3.3) 7) * : HN (Y, 71/2) = IrNY ® Z/2- rN+1,N > 2, where q is the Hopf element. Since 71 is detected by Sq 2 it is easy to see that the secondary operator in (5.3.2) is determined by the commutative diagram

bN+. HN+2Y rN+1Y (N> 2) (a)

H"(Y,1 /2)

By Spanier-Whitehead duality (see (5.2.15)) the secondary coboundary operator is given by the commutative diagram

HM-2X b"'-2- FM-1X (M> 3) (b)

HM(X,7L/2) 158 5 SPANIER-WHITEHEAD DUALITY

Hence we obtain by (5.3.2) the classical results (5.3.4) and (5.3.4)' below on the second non-vanishing groups. These results are due to J.H.C. Whitehead [CE] and Steenrod [PC] respectively and the results are dual to each other by Spanier-Whitehead duality.

For Y = Y,, N > 2, there is the natural short exact sequence (see also Theorem 15 in 8.5 of Spanier [AT])

HN(Y,71/2) (5.3.4) "7TN+1Y-HN+IY. Sg2HN+2Y

For X = X M, M > 3, we have the natural exact sequence H"'(X,Z/2) (5.3.4)' SgaHM 2Z

It remains to solve the extension problem for these sequences. Recall that for any coefficient group G the map q: G -> G ® 71/2 yields the squaring operation:

S92®G Sq2=SgG:HN(Y,G) 9'HN(Y,G®71/2) HN+2(Y,G® 71/2).

Let H; = H,(Y,71) and letH,(2) = H1(Y,71/2). For the fundamental class i = id E HN(Y, 7rNY) = Hom(HN, HN) we have the element

Sq2(i)E HN+2(Y, HN(2)) where HN(2) = HN ® 71/2 = 7rNY ® 71/2 since Y is (n - 1)-connected. For the projection p: HN(2) - cok Sq2 the universal coefficient theorem yields the commutative diagram

- Ext(HN+1,HN(2)) HN+2(Y, HN(2))-Hom(HN+2,HN(2))

1P. 1P. 1P. Ext(HN+,,cok Sq2)N HN+2(Y,cok Sq2) - Hom(HNI2,cok Sq2) where ILSg2(i) = Sq2. Thus p* ASg2(i) = 0. This shows that the element A-1p* Sg2(i) is well defined. Now the extension class of (5.3.4) is given by the formula (see Remark 2.8.8)

(5.3.5) (7rN+1Y) = A-1 p*Sg2(i) E Ext(HN+1,cok Sq2).

If we put Y = DX we get by Spanier-Whitehead duality the extension class 5 SPANIER-WHITEHEAD DUALITY 159 of (5.3.4)' as follows. We have for any coefficient group G the squaring operation

Sq2 = Sq2: HM(X,G) -- HM(X,G ® 71/2)S9z®GHM-2(X,G ® 71/2). Let H' = H'(X,71) and let H'(2) = H'(X,71/2). For the fundamental class i=idEHM(X,7TMX)=HM(X,HM)=Hom(HM,Z)®HM=Hom(HM,HM) we have the element Sg2(I) E HM-2(X, HM(2)) where HM(2) = HM ® 71/2 = ,M ® Z/2 since dim X< M. Now the universal coefficient theorem yields for the projection p: HM(2) - cok Sq2 the commutative diagram

Ext(HM- 1 HM(2))* Htif-2(X, HM(2)) µ Hom(HM-2 HM(2)) I P. IP. I 111 Ext(HM-',cok Sq2) N HM_,(X,cok Sgt)--> Hom(HM-2,cok Sgt) Here we use the fact that C*X isthe dual of C*X, thatis C*X= Hom(C*X,71), since X is a finite CW-complex. Now we have µSg2(i) = Sq2 so that p * µSg2(i) = 0. Therefore the element A-'p* Sq,(i) is well defined. Dually to (5.3.4) we get the extension class of (5.3.4)' by the formula (5.3.5)' {7.M-'X} = A-1 p*Sg2(i) E Ext(HM-',cok Sq2).

Remark The equation in (5.3.5) is due to J.H.C. Whitehead, see §18 in J.H.C. Whitehead [CE]. G.W. Whitehead reformulated this result in V.1.9 of G.W. Whitehead [RA], see also page 570 in G.W. Whitehead's book. Also Chow [SO] considers the extension problem (5.3.5). Again Shen [NH] considers the extension element (5.3.5)' which, as we have seen, is dual to (5.3.5). A proof of (5.3.5) is also given for the unstable case in Baues [CH].

Next we describe the third non-vanishing group. This group was considered by Hilton [GC] who computed the homotopy group 7rN+ 2 of an AN polyhedron, N > 2. On the other hand, Shen [NB] computed the third non-vanishing cohomotopy group. We obtain these results as follows. By the exact sequences (5.1.7) and (5.3.3)(a),(b) we get the exact sequences (Y=YN'X=XM):

bN 3 HN+3Y rN+2Y-7rN+2Y-'kerSgz (5.3.6) HM-3Xbm 3rM-2X-,7rM-2X-kerSg2 (cHM-2X), where N > 2, M > 3. Here the groups rN+ 2Y and FM-2 X depend actually 160 5 SPANIER-WHITEHEAD DUALITY only on the AN polyhedra XM_ 2where we set N = M - 2. These groups can be computed by the following result.

(53.7) TheoremThe groups rN12 Y and r M- 2X With Y = YN, X = X M and N = M - 2 > 3 are embedded in the natural diagrams (a) and (b) below in which the column and each row is a short exact sequence

HN+ IY (&Z/2 ) µ . HN+ I(Y, 7 1 / 2 ) , ° . HNY * 7L/2

(a) 'TN+ IY ®7L/2 ) > rN+2Y -a HNY *7L/2

I HN(Y,7L/2) Sg2HN+2(Y,7L/2) HM-'X (&1/2 ) ) HM-'(X,71/2) - HMX *Z/2

17. (b) 7rM-'X ®7L/2 rM-2X --+. HMX *7L/2

I HM(X,1/2) Sg2HM-2(X,71/2)

The columns are induced by (5.3.4) and (5.3.4)' respectively and the top rows are given by the universal coefficient theorem; in (b) we again use the assumption that X is a finite CW-complex. The maps q* and q* are induced by the Hopf maps. Clearly (b) is the Spanier-Whitehead dual of (a). J.H.C. Whitehead considers diagram (a) in 1.12 of [GD] and 1.18 of [NT]. The extension problems for the groups rN+2Y and ]FM-2X in (a) and (b) respectively are solved. We shall describe the extension by use of the computation of the homotopy group 'rN+ 2 M(A, n) of a Moore space.

For the computation of TrN+ 2Y and 7r M- 2X one has also to consider the secondary boundary operators bN+2 and bM-3. These operators are studied in the next result. 5 SPANIER-WHITEHEAD DUALITY 161

(53.8) TheoremLet Y = Y. and let X = X M for N = M - 2 > 3. Then the secondary boundary operators bN+3and b M-3 are embedded in the commutative diagrams (a) and (b) respectively where 4* and 4* are the secondary operations of Adem in (5.2.17).

ker SqZ c HN+3Y

I brt+3 (a) I 0.

H, (Y.Z/2) n*; a'N+2Y *HN+i(Y,/L/l) Sq2 HN+2(Y,71/2)

ker SqZ C HM 3X

146. (b) IbM3

HM(X,Z/2) 17*i FM 2X HM-'(X,1/2) Sg2HM-2(X,71/2) --a

The exact rows of these diagrams are obtained by Theorem 5.3.7. Again (b) is Spanier-Whitehead dual to (a); in fact we can assume Y = YN+3and X=XM_ 3,N = M - 3 >- 4. Then we are in the stable range so that we can apply duality. The result remains true for N = 4 by the suspension isomorphism. We see that 0* in (b) is actually the Adem operation since both and bM-3 detect the double Hopf map; see also Section 8.5. (53.9) CorollaryLet Y = YN and let X = X M and let N = M - 2 > 3. Then the Hurewicz homomorphism maps the fourth non-vanishing group surjectively to the kernel of the Adem operation: 'TN+3Y-" ker4i* CHN+3Y, 7rM-3X--kerdi*CHM 3X. We can proceed in a similar way in the discussion of the groups and operators of the F-sequences (5.1.7). The secondary boundaries bM-4 and b,,,, however, involve six different cohomology operations in the stable range. This is seen by the following spectral sequence which corresponds to the Atiyah-Hirzebruch spectral sequence for the stable cohomotopy groups; see Hilton [GC]. (53.10) Remark Let M be large and assume X is a CW-complex of dimension M, X = X M. For q < (M - 1)/2 we have the suspension isomorphism ?rM-q(X) _ [X,SM-q]= [jgX,SM' (1) 162 5 SPANIER-WHITEHEAD DUALITY which allows us to replace the cohomotopy group irM-Q(X) by the group [jqX, SM ] The latter group can be computed for q > 0 by a spectral sequence

(E,`, d,: E,'` _ E,+r'`+r+1,r> 1). (2) The E2-term is given by the cohomology groups of X

Es,'=Hs(X,Ir,(SM)) (3) with coefficients in homotopy groups of spheres. Let Ks,q c [y,q X, SM ] be the kernel of the restriction map

[lgX,SMI -> [IgxS,SM] given by the inclusion XS cX of the s-skeleton. Then we have the filtration ... cKs qc ... cKO,q=[IgX,SM].

The associated graded group of this filtration is the --term of the spectral sequence, that is Ez,q+s = KS-1,q/K5,q (4) This spectral sequence is a special case of the homotopy spectral sequence, see (III. §2) case (A) and (111.5.10) in Baues [AH]. We picture the E2-term Ez'9+S in the following diagram; by (4) the qth row in the diagram contributes to the group (1).

t-s = q

4 2 24

}bM-4

74 3 2 - -2

} bM-3

2 2 2 H:(X,x,(SM)) q2 i bM-2 2 q=I }bM-I

q=0 _ I I I I i M-4 M-3 M-2 M-1 M s The coefficients in low degrees are

i 0 1 2 3 4

7TM+i(SM) 71 1/2 1/2 71/24 0 5 SPANIER-WHITEHEAD DUALITY 163

Therefore the EZ-term yields the diagram above in which we describe the coefficients by -,2,4,24,0. The diagram indicates the possible differentials. The differentials in the row b"-q are those (higher-order) cohomology operations which are involved in the secondary boundary operator bM-q (q=0,1,2,...). For example the row bM- `in the diagram corresponds exactly to the result in (5.3.5)(b) and the row bM- 3 in the diagram corresponds to the result in Theorem (5.3.8)(b). Therefore the spectral sequence above is the precise extension of the classical results described in (5.3.2) and Theorem (5.3.8). The six differentials in the row q = 4, however, do not correspond to classical cohomology operations in the way that those in the rows q = 2 and q = 3 do. Spanier-Whitehead duality yields a dual spectral sequence consisting of homotopy groups and homology operations which can be used for the computation of stable homotopy groups of a finite CW-complex; this extends the duality discussed in (5.3.2) and Theorem (5.3.8).

5.4 The stable CW-tower and its dual

Let N < M < 2 N - 1. The stable CW-tower is the restriction of the CW-tower of categories to the subcategory of CW-complexes X = XN with dim X:5 M and trivial (N - 1)-skeleton XN-' = *. The stable CW-tower is defined by homotopy systems which consist of a chain complex C and a CW-complex X" which realizes C in degree _< n. We now consider dually cohomotopy systems which consist of a cochain complex D and a CW-complex Y" which realizes D in degree >- n. Cohomotopy systems yield a tower of categories which is Spanier-Whitehead dual to the stable CW-tower. We fix N and M as above. Let AN - N be the full homotopy category of finite CW-complexes of the form X = XN ; see Section 5.2

(5.4.1) Definition We define an (N, M)-homotopy system of order (q + 1) and an (N, M)-cohomotopy system of order (q + 1) to be triples

(C, f,XsN,+q),resp.(D,g,Yq) with the following property (a), resp. (a') where (a') is the dual of (a). As usual let C* and C* be the reduced cellular chain complex and cochain complex respectively.

(a) C = (C, d) is a chain complex of finitely generated free abelian groups with C; = 0 for i < N and i > M and C coincides with C * XsN, + qin degree 5 N + q. Moreover

C N+q f:N+,+1 N+qX 164 5 SPANIER-WHITEHEAD DUALITY

is a homorphism such that fd = 0 and such that

d: CN+q+I ITN+qXN+q h HN+qXN+q CCN+q

is the differential of C. (a') D = (D, d) is a cochain complex of finitely generated free abelian groups with Di = 0 for i < N and i > M and D coincides with C* YMq in degree >_ M - q. Moreover, g: DM-q- I , .M-qyM q

is a homomorphism such that gd = 0 such that

DM-q- I_-g--), .M-qYM d: q -LHM-qY,yqCDM-q

is the differential of D.

We obtain maps

(b) (r,n): (C, f, XN+q) - (C', F', UN +q) (b') (D',g',VMq) - (D,g,YM-q) between such homotopy systems, resp. such cohomotopy systems, as follows. In (b) the map : C - C' is a chain map and 71: XN+q -> UN+q is a map in 0 CW/= such thatt: coincides with C * rl in degree < N + q and such that f'Z: = 7.f on CN+q+1. In (b') the map t : D - D' is a cochain map and : VM q YMqis a map o in CW/ such that 6 coincides with C*-q in degree >_ M - q and such that g'6= -1*g on DM-q- I. Let H(q+l) and H(q+1) be the category of such homotopy systems and cohomotopy systems respectively.

Remark An (N, M)-homotopy system of order (q + 1) is a special homotopy system of order N + q + 1 in the sense of Chapter 4. We recall above the definition of homotopy system so that the duality between homotopy systems and cohomotopy systems becomes evident.

(5.4.2) Definition We define the homotopy relation = on H(q). For maps as in (b) we set (1,,1) = (, i) if there exist homomorphisms aj +:CC -+ Ci'+ 1, j >_ N + q, such that inthe abelian group [XN+q, UN+q] we have the equation IV - {77) =p*(f'aN+q+1) 5 SPANIER-WHITEHEAD DUALITY 165 where p: XN+q XN+q is the quotient map and where p* is defined on

Hom(CN+q,1TN+q UN +q) = [ XN+q , UN+q compare the convention in (5.1.2). Moreover:

Sk - 6k = akdk +dk+Iak+I,k> N+q. Dually we define the homotopy relation = on H(q). For maps as in (b') we set i) = il) if there exist homomorphisms

ai-I: D'- (D')i-I 1

('n) - (71) _ i*(g(aM-q-I )). Here is X1 q -> XM -q is the inclusion and i * is defined on

Hom(DM 9, irM 4(VM q)) =[Vq' XM-q J . Moreover, k-0= ak d k +dk- lak-k

A: H(q+1) H(q)I (C, f, XN+q) - (C, f', XN+q- (5.4.3) A: Hcq+n -* H(q), (D,8,YM q) (D, 8', 1'N q+,). Here f is the attaching map of (N + q)-cells in XN+q and g'is the coattaching map of (M - q)-cells in YM q; see (5.1.5). These functors induce functors between the corresponding homotopy categories. We have obvious isomorphisms of categories

HOII = FChainN-N(covariant), (5.4.4) H' )= FCochainN-N(contravariant).

Here FChainN - N is the category of finitely generated free chain complexes C with C, = 0 for i < N and i > M. Similarly FCochainN N is the category of finitely generated free cochain complexes D with D' = 0 for i < N and i > M. 166 5 SPANIER-WHITEHEAD DUALITY

The next theorem describes the stable CW-tower and its dual. The stable CW-tower is just the restriction of the CW-tower in Section 4.3 to the stable subcategory AN-N; the new featureis the dual of the stable CW-tower obtained in this theorem.

(5.4.5) TheoremLet N < M < 2 N - 1. The categories H(q) form a tower of categories. Dually the categories H(q) form a tower of categories. Both towers approximated the homotopy category AN - N. Moreover Spanier-Whitehead dual ity yields a contravariant isomorphism between these towers of categories as indicated in the following diagram. D AM-N AM-N N N

1 1 H(M-N+l) H(.M-N+ l)

HN+qrN+q-----?----.aHM-qFM q

HN+q+lrN+q----T -;----> HN-q-lrM-q HO) H(1)

11 II. FChainN-N/= FCochainN N/=

The Spanier-Whitehead duality isomorphism D on AN-' is defined in (5.2.1). We now define D on H(q+!) by

(5.4.6)

DC is given by (DC)M-q = CN+q. Thus b is a covariant functor by (Dr )M-q=N+q We define D on maps Q,71) by 5 SPANIER-WHITEHEAD DUALITY 167

DQ, 71) = (De, D-q). The properties of Spanier-Whitehead duality in Section 5.2 show that D is a well-defined contravariant isomorphism

D: H = H(q+1) of categories. We obtain the obstruction operator ® on H(q) and the action + on H(q+1) in the same but dual way as in Section 4.3. For this we use the H(q )-bimodules H,_ eI''° with m = M - q, e > 0, which are defined by the homology groups

(5.4.7) (Hm_eI''")(V,Y)=H,,,-e(D*,FmV) where Y = (D, g, YZ) and V = (D', g', V,M) are objects as in Definition 5.4.1(b). The chain complex D* = Hom(D,7L) is the dual of the cochain complex D and Im(V) = Fm(V) is defined as in (5.1.8) by the space V which realizes the coattaching map g. Hence (5.4.7) is contravariant in V and covariant in Y. The natural isomorphism T in the diagram of Theorem 5.4.5 is given by (X,YE H(q))

T (5.4.8) HM-q-e (DX, rM-qDY) HN+q+e(X rN+qV) where we use (5.2.10) and (5.2.13).

Using Theorem 5.4.5 each result on the stable CW-tower corresponds to a dual result for the dual tower. In particular the results on boundary invariants in Chapter 4 have interesting dual formulations. 6 EILENBERG-MAC LANE AND MOORE FUNCTORS A EILENBERG-MAC LANE FUNCTORS

We consider three types of Eilenberg-Mac Lane functors given by homology, cohomology, and pseudo-homology of Eilenberg-Mac Lane spaces. While the homology and cohomology of Eilenberg-Mac Lane spaces is extensively studied (see Eilenberg and Mac Lane [I], [II], Cartan [HC], and Decker [IH]), the pseudo-homology of Eilenberg-Mac Lane spaces is not treated in the literature. In our theory of boundary invariants, however, the pseudo- homology arises naturally in the same way as the cohomology in the theory of k-invariants. For this reason we describe the cohomology and pseudo- homology of Eilenberg-Mac Lane spaces along parallel lines. We classify (m - 1)-connected (n + 1)-dimensional homotopy types X with ir,(X)=0for m

Such homotopy types X have an (n - 1)-type ,(X) whichis an Eilenberg-Mac Lane space. Therefore our classification theorem (Section 3.4) yields explicit algebraic models of such homotopy types in terms of the Eilenberg-Mac Lane functors.

6.1 Homology of Eilenberg-Mac Lane spaces

For m > 2 the homotopy category typeset of (m - 1)-connected m-types is equivalent to the category Ab of abelian groups. In fact, each (m - 1)- connected m-type X is an Eilenberg-Mac Lane space and one has the equivalence of categories

(6.1.1) Ab -4 typeset which carries the abelian group A to the space K(A, m). The inverse of this equivalence carries X to the abelian group Trm(X). Using this equivalence of categories we identify homomorphisms A -- B in Ab with homotopy classes of maps K(A, m) - K(B, m), that is

Hom(A, B) = [K(A, m), K(B, m)]. (1) 6A EILENBERG-MAC LANE FUNCTORS 169

In particular, the homomorphism µ: A X A -A given by addition in A (with µ(x, y) = x + y for x, y E A) yields up to homotopy a map µ: K(A, m) X K(A, m) = K(A xA, m) -. K(A, m) (2) which gives K(A, m) the structure of a homotopy commutative H-space. The multiplication µ can also be obtained by loop addition since we have a canonical homotopy equivalence K(A, m) = 0 K(A, m + l) (3) where SOX denotes the loop space of X. The loop space functor fl: types'+ i -'types' (4) is an equivalence of categories compatible with the equivalence (6.1.1).

Remark We point out that the equivalence of categories in (6.1.1) is actually induced by a functorial construction of the space K(A, m) as follows. For a topological monoid M let B(M) be the classifying space of M; the space B(M) is obtained as the realization of a simplicial space as for example in Baues [GL]. If M is abelian then B(M) turns out to be again an abelian topological monoid in a canonical way so that in this case iteration is possible. Hence one obtains a functorial construction of the Eilenberg-Mac Lane space by the m-fold iterated classifying space K(A, m) = BB . . . B(A). Here the abelian group A is an abelian topological monoid with the discrete topology. Compare also Segal [CC].

(6.1.2) DefinitionThe (classical) Eilenberg-Mac Lane functor m): Ab -> Ab is the composite H Ab -> typeset Ab of the equivalence (6.1.1) and the homology functor H of degree n. Hence the functor m) carries the abelian group A to the homology group H (A, m) = H (K(A, m)) of the Eilenberg-Mac Lane space K(A, m). The total homology H(A, m) = H,, (K(A, m)) is a graded commutative ring via the multiplication H(A, m) ®H(A, m) H. (K(A, m) x K(A, m)) H(A, m). Here we use the cross-product in homology and the multiplication µ in (6.1.1X2).

(6.13) Definition We also define Eilenberg-Mac Lane bifunctors

Him>, Ab°P X Ab - Ab 170 6A EILENBERG-MAC LANE FUNCTORS be the cohomology H(" ,,)(A, B) = H"(K(A, m), B) and the pseudo-homology H (B, K(A, m)) respectively. Here B is the group of coefficients.

The universal coefficient sequences yield short exact sequences

(6.1.4)

/A (6.1.5)Ext(B, H m)) which are natural in A, B E Ab.

(6.1.6) PropositionThe functor H +' is, by (6.1.4), a kype functor on Ab and the functor H,(,'") is, by (6.1.5), a bype functor on Ab. Moreover H(n.+) 'is dual to in particular the kype functor H(,n) 1is split if and only if the hype functor is split.

Proof Let K* : Ab -b Chainz/= be the functor which carries an abelian group A to the singular chain complex K,, (A) = C* K(A, m). Then we have H(nm)'(A,B)=H"+ 1(K*(A),B)

H,(,m'(B, A) = K,, (A)). Hence the proposition is a consequence of Theorem 3.3.9.

The next result is due to Decker [IH].

(6.1.7) TheoremLet k < m for m odd, and for m even let k:5 m. Then the kype functor H(m j k is split.

Using Proposition 6.1.6 we get the corresponding result for the dual bype functor:

(6.1.8) TheoremLet k < m - 1 form odd, and form even let k:5 m - 1. Then the hype functor Hmm )k is split.

6.2 Some functors for abelian groups

In order to give explicit descriptions of some Eilenberg-Mac Lane functors we have to introduce various basic functors and constructions on abelian groups, some of which are quite bizarre. We start with the classical torsion product. 6A EILENBERG-MAC LANE FUNCTORS 171

(6.2.1) DefinitionThe torsion product is a functor Tor: Ab X Ab --> Ab. For abelian groups A and B let Tor(A, B) be the abelian group whose generators are the symbols Th(a, b) for all positive integers h and all pairs of elements a E A, b E B such that ha = 0 and hb = 0. These generators are subject to the relations Th(a,b,+b2)=Th(a,b,)+Th(a,b,), ha=0=hb1=hb2 Th(a,+a2,b)= rh(a,,b)+Th(a2,b), hat=ha,=0=hb

Thk(a, b) = rh(ka, b), hka = 0 = hb

Thk(a, b) = r,(a, kb), ha = 0 = hkb. The torsion product, so defined by Eilenberg and Mac Lane [II], agrees with the usual definition derived from the tensor product of abelian groups. For this we choose a short exact sequence (6.2.2) 0-9R-F->A ->0 where F and hence R are free abelian. We call d = dA a short free resolution of A. Then A * B is the kernel of d ® B in the exact sequence 0->A*B-- R®B- F®B->A®B->0. This kernel is independent, up to a canonical isomorphism, of the choice of the short free resolution dA. In particular we choose F = 7L[A] to be the free abelian group with generators [a], for all a E A, and R to be the subgroup of F generated by all [a,] - [a, + a,] + [a2], for a,, a2 EA. Then the correspondence [a] -a induces an isomorphism F/RA. Using this standard resolution of A we obtain Tor(A,B)-A*BcR®B (6.2.3) Th(a, b) H (h[a]) ® b. To justify the definition of the isomorphism we observe that h[a] E R since ha = 0 in A and that (d ® B)(h[a]) ® b = h[a] ® b = [a] ® hb = 0 since hb = 0. For a further discussion of the binatural isomorphism (6.2.3) compare 11.3 in Eilenberg and Mac Lane [II]. Next we consider functors which we derive from Whitehead's I'-functor and the exterior square A2 in Section 1.2. We have natural homomorphisms r(A)"A®A I'(A) (6.2.4) A2(A)-A®A-iA2(A). 172 6A EILENBERG-MAC LANE FUNCTORS

Here H in the top row is defined by Hy(a) = a ® a and P is the Whitehead product defined by P(a ®b) = [a, b] = y(a + b) - y(a) - y(b) where y: A - IF(A) is the universal quadratic map. On the other hand, H in the bottom row is given by H(a A b) = a ® b - b ® a and we set P(a (& b) = a Ab. The homomorphisms H, P satisfy PHP=2P and HPH=2H so that (6.2.4) describes quadratic 77-modules in the sense of Definition 6.13.5 below. Recall that Chainz denotes the category of chain complexes C._ {C", d"}"E rA cochain complex C* = {C", d ,l,,E zis identified with the chain complex C,, given by C" = C-", d,, = d-".

(6.2.5) DefinitionWhitehead's quadratic functor IF and the exterior square A2 induce chain functors

r,,,M. : Ab -* Chainz/= (1) as follows. We choose for each abelian group A a short free resolution d denoted by d = dA: X1 -> X0. Then we define for F = I' or F = A2 the chain complex F. WA) by

F2(dA) F,(dA) FQ(dA)

(2) II II II sF(X,)z X1 ®X1 ®X, ®Xo s' F(Xo)

31 = (F(dA), P(dA (9 X0)), 52=(P,-XI0 dA). The chain functors F* = IF,, M. are examples of the quadratic chain functors in Definition 6.14.3 below. We consider dA as a chain complex concentrated in only two degrees. The homology of dA is the abelian group A. For a homomorphism gyp: A - B we can choose a chain map

dA -->dB, which induces P in homology. This chain map induces F*(dA)---,F*(dB) given by (qpo, cp,)* (p, 0 cpo, 91 ®9,). The homotopy class of (qpo, (PI) *depends only on the homomorphismgyp. Hence we obtain well-defined functors F* and AA*in (1) which carry A to F*(dA), resp. A2*(dA) and which carry a homomorphism 9 to the homotopy class of (cPo,PI)*- 6A EILENBERG-MAC LANE FUNCTORS 173

We now consider the homology of the chain functors. As usual we define for a chain complex C * = (C,,, d"), resp. cochain complex C (C", d ") the homology groups H"(C*) = kernel(d")/image(d"+,) H"(C*) = kernel(d")/image(d"-1)

We leave it to the reader to show

(6.2.6) PropositionOne has Her*(dA) = 0 and H2AA*(dA) = 0. Moreover one has natural isomorphisms r(A) = Hor*(dA) and A2(A) = HO AZ. WA).

The remaining homology is used in the following definition.

(6.2.7) DefinitionUsing the chain functor in Definition 6.2.5 we define the torsion functors rT,A2T:Ab-*Ab by I'T(A) =H,I'*(dA) and AZT(A) = H,A2,k(dA). These are quadratic functors with cross-effect rT(A I B) =A * B = AZT(A I B) given by the torsion product in Definition 6.2.1. Compare Section 6.13 below.

(6.2.8) Remark Using the notation in Section 6.14 we have FT(A) =A *,Zr and AZT(A) =A *71A so that the cross-effect in Definition 6.2.7 are obtained by 7.7 in Baues [QF].

The torsion functors above yield an interpretation of the bizarre functors fl and R introduced by Eilenberg and Mac Lane [II], §13 and §22.

(6.2.9) Theorem One has a natural isomorphism rT(A) = R(A).

Proof We here recall the definition of the functors R and we define the isomorphism in terms of generators. Let R(A) denote the quotient group

R(A) = Tor(A, A) ®r(2A)/L5(A) (1) with 2 A = kernel(2: A --*A) =1 /2 * A. Here L5(A) is the subgroup generated by the relations (h E 7L, a, s, t E A)

T,, (a,a)=0 ha=0 (2) [s, t ] = T2(s, t) 2s = 2t = 0 (3) where [s, t] = y(s + t) - y(s) - y(t) E r(2A) is the Whitehead product. We 174 6A EILENBERG-MAC LANE FUNCTORS construct special cycles (4) and (5) below in T,(dA). Let a, b, x E A and let dA be the standard resolution so that X0 = Z[ A] is freely generated by elements [a], a E A. We then define for ha = hb = 0 the element

0h(a, b) _ (h[a]) ®[b] - (h[b]) ®[a] E F,(dA) (4) where h[ah[b] EX, cX0. The element (4) is obviously a cycle, that is 518h(a, b) = 0. Moreover we obtain for 2x = 0 the element

02(x) = y(2[x]) - (2[x]) ® [x] E I',(dA) (5) which is a cycle since y(2[x]) = 4y([x]) = 2[[x],[x]] in I'(X0). Using the elements (4) and (5) we define the isomorphism

0: R(A) - FT(A) (6) by O{-rh(a, b)) = (0h(a, b)) and 0(y(x)) = {02(x)}, see (1). We study the I'-torsion fT(A) in more detail in Section 11.2.

(6.2.10) Theorem One has a natural isomorphism AZT(A) = [1(A).

Proof We recall the definition of 11(A); see Eilenberg and Mac Lane [II], §13. We define the group 11(A) to be the abelian group generated by the symbols wh(x), for positive integers h and elements x EA with hx = 0, subject to the relations

Whk(x) = kwh(x) hit = 0 (1)

kwhk(x) = wk(kx) hkx = 0 (2)

wh(kxly)=whk(xly) hkx=hy=0 (3) Wh(XIyIZ)=0 hx=by=hz=0. (4) Here we use for a function f: A -' B the notation

f(xIy)=f(x+y)-f(x)-f(y), (5) f(xlylZ) =f(x+y+z) -f(x+y) -f(x+Z) -f(y+Z) +f(x) +f(y) +f(z).(6) Hence f(x I y) is bilinear in x, y if and only if f(xl yl z) = 0. Now let dA: X, -> Xo be the standard resolution of A so that Xo = l[ A] is freely generated by elements [a], a EA. We then define for x EA with hx = 0 the element Oh(x) _ (h[x]) ®[x] EXI ®XO cA;(dA) (7) 6A EILENBERG-MAC LANE FUNCTORS 175 where h[x] EX, cX0. One readily checks that ©h(x) is a cycle, that is S,©h,(x) = 0. Using the element (7) we define the isomorphism

O: f2(A) - AZT(A) (8) by ©(wk(x)) = {Oh(x)}. For a cyclic group 7L/m7L, m > 1, generated by x we obtain the cyclic group

fl(Z/mZ) = 7L/m7L generated by wm(x).

We now use the chain functors for the definition of torsion bifunctors. For this recall that [C*,K*] denotes the set of homotopy classes of chain maps C * - K*. For an abelian group B let

dB: X, -+X0, sdB: Y2 -> Y, be short free resolutions of B considered as chain complexes.

(6.2.11) Definition We introduce the torsion bifunctors

FT*,FT*,AZT*,AZT*: Ab°P xAb ->Ab.

Here we use for F = t or F = AZ the pseudo-homology and cohomology of F. dA with coefficients in B which defines: FT*(B,A) = [dB,F*dA]=Ho(B,F*dA) FT*(A,B) = [F*dA,sdB] =H'(F*dA, B).

Using the universal coefficient sequences one has the following natural short exact sequences.

Ext(B,FT(A)) >-> FT*(B, A) -. Hom(B, F(A)) Hom(fT(A),B) Ext(B, AZT(A)) >-> AZT*(B, A) - Hom(B, AZ(A)) Ext(AZ(A), B) >-> AZT*(A, B) -. Hom(AZT(A), B). Hence FT* is a bype functor dual to the kype functor FT# and AZT* is a bype functor dual to the kype functor AZT*. These functors are linear in B and quadratic in A; see Section 6.13 below.

Next we describe a further pair of bifunctors L*, L* which are dual to each other. 176 6A EILENBERG-MAC LANE FUNCTORS

(6.2.12) DefinitionAs in Eilenberg and Mac Lane [II] §27 one obtains a bifunctor L#:Ab°PxAb- Ab as follows. Let L#(A, B) be the abelian group consisting of all pairs (a, b) where b: AZ(A) -> B is a homomorphism and where a: A -> B ® 2/2 is a function satisfying the condition a(x+y)-a(x)-a(y)=b(xAy)®1 for x, y E A. One has the natural short exact sequence

Ext(A®71/2, B) ° L#(A,B)Hom(M(A),B) with µ(a, b) = b and A(a) = (Oa, 0) where 0: Ext(A ® 2/2, B) = Hom(A (9 Z/2, B ® 71/2) = Hom(A, B ® Z/2). Hence L# is a kype functor. There is a different definition of L# by use of the `quadratic Hom functor' in Definition 6.13.14 below. For this we use the `quadratic 2-module' L'(B)=(71/2(9 B-0-4B-2-1 2/2®B) where q is the quotient map. Then we have a canonical binatural isomorphism L#(A, B) = Hom7(A, L'(B)).

(6.2.13) (A) Definition We introduce the bifunctor L#: Ab°P x Ab - Ab as follows. If A and B are finitely generated let L#(A, B) be the abelian group generated by elements (b, a) and (,8) with b E B, a E Hom(A, 71/2), and 19 E Ext(A, A2(B)). The relations are (/3+P')=(0)+(/3') (b, a+a')=(b,a)+(b,a') (b+b',a)= (b, a)+(b',a)+(bAb')*(da). Here d: Hom(A, Z/2) -> Ext(A/2) is the natural connecting homomorphism induced by the short exact sequence 71 2 - 71/2. Moreover (b A Y).: Ext(A,71) - Ext(A, AZB) is induced by the homomorphism 2 -* A2B which 6A EILENBERG-MAC LANE FUNCTORS 177 carries 1 to b A b'. It is obvious how to define induced maps for the bifunctor L.. One has the natural short exact sequence

Ext(A, A2B) N L#(A, B) Hom(A, B (9 71/2) with 0(/3) _ 0 and A(,6) = 0, µ(b, a) = b*(a) where b*is induced by the homomorphism Z/2 - B ® 71/2 which carries the generator 1 to b 0 1. The exact sequence shows that L. is a bype functor on Ab. The definition of L. can also be obtained by the quadratic tensor product in Definition 6.13.13. In fact, we have the binatural isomorphism L#(A, B) = B ®z L(A) where L(A) = (Hom(A,71/2) - Ext(A,71) -L Hom(A,71/2)). Here d is the connecting homomorphism induced by the exact sequence Z-z->Z-'71/2.

(6.2.13) (B) DefinitionFor arbitrary abelian groups A and B we have to use the following more intricate definition of the bifunctor L,, above. Let 71[B] be the free abelian group generated by the set B. We have the following commutative diagram with short exact rows

i P2, K2(B) > 71[B] ®71/2 B ® 71/2

IYB A2(B)®71/2r=_1'_1r(B)®71/2 °OB®71/2 compare Section 1.2. Here K2(B) is the kernel of the canonical projection P2 with p2([b] ® 1) = b ® 1 and y'isthe homomorphism defined by y'([b] (9 1) = (yb) ® 1 for b c- B. Using the natural transformation yB we obtain the following push-out diagram Hom(A,K2(B)) N Hom(A,71[B] (&71/2) Yl push I Ext(A, A2B) >' L4,(A, B) 00 Hom(A, B ® 71/2) where y(a) = a*ye with yB E Hom(KB, AZ(B) (9 Z/2) = Ext(KB, AZ(B)) as above. This completes the definition of the bifunctor L, in Definition 6.2.13(A). One can check that Definition 6.2.13(A) coincides with the one here since we have the isomorphism of quadratic 11-modules L(71/2) = (71/2 _1 # 71/2 -0 Z/2) =Zr® 71/2 so that B 0 L(Z/2) = r(B) 0 71/2. 178 6A EILENBERG-MAC LANE FUNCTORS

(61.14) TheoremThe functors Lv and L* above are dual to each other. Proof Our proof is indirect. In fact, Eilenberg and Mac Lane [II] show H(63)(A, B) = L*(A, B) ®Hom(A *7L/2, B). On the other hand, we show below H53)(B, A) = L4,(B, A) ® Ext(B, A *Z/2). This implies that L" and L,1, are dual to each other since we know that H(3) and H531 are dual to each other; see Proposition 6.1.6.

6.3 Examples of Eilenberg-Mac Lane functors

Many cases of Eilenberg-Mac Lane functors are computed explicitly in the literature. Compare Eilenberg and Mac Lane [II], Cartan [HC], Serre [CM], and Decker [IH] for the computation of the groups H,,(A,m) =H"(K(A,m)), H(m)(A, B) = H"(K(A, m), B). The pseudo-homology H,1,m)(B, A) =H"(B, K(A,m)) is not treated in the literature. By the work of Cartan one has a small model of the chain complex C* K(A, m) which can be used for the computation of A) as a functor in A and B. On the other hand, we can use the duality of Proposition 6.1.6 which shows that the functors H(mj 1 and determine each other; see Section 3.3. The duality, however, does not give us an appropriate formula for Him) if we know such a formula for H(' n' )1 or vice versa. We now describe, for small values of m and r = n - m, some of the Eilenberg-Mac Lane functors above. For the classical Eilenberg-Mac Lane functor Hm+,(-, m) one has the following list of natural isomorphisms, AEAb,m>2. (6.3.1) Hm(A,m)=A and Hm+1(A,m)=0

(6.3.2) Hm+z(A,m)=(A(®7L/2m>-3. Here Iis Whitehead's functor (see Section 1.2). FT(A) m=2 (6.3.3) Hm+3(A,m)= A2(A)®A*Z/2 m=3 A*7L/2 mz4. 6A EILENBERG-MAC LANE FUNCTORS 179

Here A * B is the torsion product of abelian groups and AZ(A) is the exterior square. Moreover FT(A) is the F-torsion of A (see Section 6.2); this is R(A) in the notation of Eilenberg and Mac Lane [II]; see Theorem 6.2.9.

I'6(A) m=2 3 AZT(A)®A®71/3 m=3 (6.3.4) Hm+4(A,m)= I'(A)®A®71/3 m=4 A(9 (71/2®71/3) m;-> 5.

Here A2T(A) is a AZ-torsion of A (see Section 6.2); this is fl(A) in the notation of Eilenberg and Mac Lane [II]; see Theorem 6.2.10. Moreover I76(A) is part of the free algebra with divided powers IF,, A generated by A; see Eilenberg and Mac Lane [II].

see Decker [IH] m=2 AOA ®71/2®A®Z/3 m=3 (6.3.5) Hm+s(A, m) = I'T(A) ®A *71/3 m=4 AZ(A) ®A * (71/2 ® 71/3)m=5 A*(71/2(D 71/3) m>_6.

The results of (6.3.1)-(6.3.5) were essentially obtained by Eilenberg and Mac Lane [II]. One can find further explicit functorial descriptions of Hm+,(A, m) in Decker [IH], in particular, in the metastable range r < 2m. Next we consider for r< 4 the bifunctors

(6.3.6) H(m) +.+', H(, ,'"+)r: Ab°P x Ab-+ Ab which are dual to each other with kype and bype structure given by (6.1.5) and (6.1.4) respectively. Hence we have in the split case, see for example Theorems 6.1.7 and 6.1.8, the natural isomorphisms

(6.3.7) Hcmj'+'(A, B) = Ext(H,(A, m), B) ®Hom(Hm+,+(A, m), B) Ext(B,Hm+r+i(A,m)) ® Hom(B,Hm+,(A,m)) where the right-hand side is determined by the functors in (6.3.1)-(6.3.5). We now describe formulas for the functors in (6.3.6) in the following lists, where split means that one has to apply formula (6.3.7); moreover 'ED split' means that one has to use formulas as in (6.3.7) for the remaining terms described in 180 6A EILENBERG-MAC LANE FUNCTORS the same line; compare the example on H(5)10and H95) in (6.3.12) below. (63.8)

H(m+ 2(A, B) Hm(m+)I(B, A) Hm+,(A, m) Hm+2(A, m) m=2Hom(I'A,B) Ext(B,FA) 0 f(A) m3Hom(A®7L/2, B)Ext(B,A®7L/2) 0 A®Z/2

(63.9) H(mj 3(A, B) A) Hm+2(A, m) Hm+3(A, m) M = 2 rT#(A, B) rT#(B, A) F(A) FT(A) m=3 L#(A,B)® L#(B,A)® A®7L/2 A2(A)® Hom(A *7L/2, B)Ext(B, A *7L/2) A *7L/2 m>4 4 split split A ® 71/2 A *Z12 For the definition of the torsion bifunctors FT* and FT#, see Definition 6.2.11 and for the definition of L#, L. see Definition 6.2.13 and Theorem 6.2.14.

(63.10) H(mj 4(A, B) Hm+)3(B, A)Hm+3(A, m) Hm+4(A, m) m=2 FT(A) T6(A) m=3 A2T#(A, B) A2T#(B, A) A2(A) A2T(A) ® split ® split ®A * 71/2 ®A ® 71/3 m=4 split split A *7/2 T(A) ®A ® 71/3 m >_ 15 split split A *7/2 A ® 71/6 The split cases are consequences of Theorems 6.1.7 and 6.1.8. (63.11) H(m+)5(A, B) A)Hm+4(A, m) Hm+5(A, m) m=2 r'6(A) see Decker [IH] m=3 A2T(A) A®A®7L/2 ®A ® 71/3 ®A *Z/3 m = 4 1'T#(A, B) FT#(B, A) T(A) FT(A) ® split ® split ®A 0 Z/3 ®A * Z/3 m= 5 L#(A, B) L#(B, A) A® 71/2 A2(A) ® split ® split ®A o 2/3 ®A * Z/6 m >_ 6 split split A 0 71/6 A *Z/6 6A EILENBERG-MAC LANE FUNCTORS 181

The split case m >_ 6 is again a consequence of Theorem 6.1.7 and 6.1.8. For example the case m = 5 in the list means that one has natural isomorphisms of kype functors and bype functors respectively:

(6.3.12) H(S0)(A, B) = L#(A, B) ®Ext(A (9 71/3, B) ®Hom(A *Z/6, B), H95)(B, A) = L#(B, A) ® Ext(B, A *71/6) ® Hom(B, A (9 1/3).

For the definition of L# and L# see Definitions 6.2.12 and 6.2.13.

Proof for the lists (6.3.8)-(6.3.11)One finds all computations of the cohomology H°'+,+1 forr< 4 in §27 of Eilenberg and Mac Lane [II]. The case HHj is not treated by Eilenberg and Mac Lane. This case is obtained as follows. We shall show in Theorem 9.4.2 that H42) = FT#. Since FT* is dual to FT# and since H(2) is dual to H42) this also implies H(2) = FT #. Hence we use here duality in a crucial way. We also use duality for the computation of For example H150 in (6.3.11), m = 5, is known by Eilenberg and Mac Lane [II]. This yields H95> since we have the duality of L# and L. in Theorem 6.2.14. The cases H(3), H63), H(4), and H84 are treated in the Diplomarbeit of J. Wendelken [KEM].

6.4 On (m - 1)-connected (n + 1)-dimensional homotopy types with7r;X=0 for m

We consider homotopy types of (n + 1)-dimensional spaces X for which the (n - 1)-type P -,(X) is an Eilenberg-Mac Lane space K(A, m), m > 2. For such homotopy types the classification theorem in Chapter 3 can be applied effectively since the category C in this case is just the category of (m - 1)- connected m-types which is equivalent to the category of abelian groups. Moreover the bype and kype functors in question are the Eilenberg-Mac Lane bifunctors discussed in Section 6.1; explicit examples of Eilenberg-Mac Lane functors are described in Section 6.3 above. Let spaces(m, n), be the full homotopy category of (m - 1)-connected (n + 1)-dimensional CW-spaces X with Tri(X) = 0 for m < i < n. Moreover let types(m, n), be the full homotopy category of (m - 1)-connected n-types Y with iri(Y) = 0 for m < i < n. Hence Y has at most two non-trivial homotopy groups aY and ir,,Y. Recall that we have the Eilenberg-Mac Lane functors H(m+) 'and H,1,')in Sections 6.1 and 6.3 which are kype functors and bype functors respectively. The next result is an immediate application of the classification theorem 3.4.4; it gives us explicit algebraic models of homotopy types. 182 6A EILENBERG-MAC LANE FUNCTORS

(6.4.1) Classification theoremLet 2< m < n. Then one has detecting functors

A : spaces(m, n) . -> Kypes(Ab, H('mj'

A': spaces(m, n),- Bypes(Ab, H,m') A: types(m, n),rkypes(Ab, H( ') = Gro(H(,,,)'

A': types(m, n),r -> bypes(Ab, H,,,'")).

Here the categories on the right-hand side are the purely algebraic categories given by the kype functor H(,,,)' and the bype functor H, m ;see Sections 3.1 and 3.2. The detecting functor A in the classification theorem is well known in the literature. The more sophisticated detecting functors A, A', and A', however, yield new results on the algebraic classification of homotopy types.

Proof of Theorem 6.4.1Consider Theorem 3.4.4 where we take m + r = n and C = typesO - Ab. (*)

Then we have

spaces'm ' (C) = spaces(m, n) typesm(C) = types(m, n) and E, resp. F, in Theorem 3.4.4 coincide with H(m 1, resp. Him), by use of the equivalence (*). This immediately yields the proposition of Theorem 6.4.1. 11

Recall that the used in Theorem 6.4.1 are tuples X=(A,ir,k,H,b) where A, zr, H are abelian groups and

kEH(1'I(A,ir) (6.4.2) b E

'_ H- I(A, m) it is exact. The kype X is free if H is free abelian; then X is an object in Kypes(Ab, H(,,,+)1) and thus X determines via A in Theorem 6.4.1a unique homotopy type X in 6A EILENBERG-MAC LANE FUNCTORS 183 spaces(m, n)" with A(X) _- X. In (3.1.5) we associate with X the exact r-sequence which is the top row in the commutative diagram, A = lrm X,

(6.4.3)

H H"+i(A,m)µ(k) IT - H(kt)->H"(A,m)--0

1 -- II II II II H"+,X ---> r,,X - vr"XH"X -I'" _,X -' 0 which is a natural `weak' isomorphism of exact sequences. The bottom row is part of Whitehead's certain exact sequence for X; compare Theorem 3.4.4. In particular we get:

(6.4.4) TheoremLet X be an (m - 1)-connected space, m >: 2, with Tr;X = 0 for m < i < n. Then the homology H,, X is determined as an abelian group by the k-invariant k e H(m '(7rm X, ir,, X) of X since we have the isomorphism H,, X H(kt) in (6.4.3).

Remark The main theorem of Eilenberg and Mac Lane [RH] shows that the cohomology H"(X, G) of a space X as in (6.4.4) is determined up to isomorphism by the k-invariant k of X. Their description of H"(X,G) relies on a choice of a cocycle representing k. The direct computation of H"(X) = H(kt) in terms of k in Theorem 6.4.4, however, was not achieved and seems to be new; compare the remark following the theorem of Postnikov invariants in Theorem 2.5.10. Using Theorem 6.4.4 we obtain H"(X,G) as an abelian group by

H"(X,G) = Ext(H,,_,(ir"X, m),G) ® Hom(H(kt),G) where we use the universal coefficient formula.

On the other hand, recall that the H,,(')-bypes in Theorem 6.4.1 are tuples

(6.4.5)

Here A, H0, H, are abelian groups and

(b E Hom(H1, H"+,(A,m)),

S10 E H,(,m)(H0, A, b) = cok(D Ext(Ho, b)),

A Ext(H0, b): Ext(Ho, H,) -> Ext(Ho, H,,, (A, m)) -'H,(,m'(H0, A), such that µ( j3): Ho -+ H,,(A, m) is surjective. The bype X is free if H, is free 184 6A EILENBERG-MAC LANE FUNCTORS abelian; then X is an object in Bypes(Ab, and thus X determines via A' in Theorem 6.4.1 a unique homotopy type in X in spaces(m, n) with A'(X) - X. In (3.2.5) we associate with X the exact f-sequence which is the top row in the commutative diagram, A = Hm X,

(6.4.6) H1 b Tr(pt)-> Ho

II b_IX II 1 II II F,,-1X->0.

This is a natural `weak' isomorphism of exact sequences; compare Theorem 3.4.4. Dually to Theorem 6.4.4 we thus get:

(6.4.7) TheoremLet X be an (m - 1)-connected space, m >_ 2, with 7ri(X) = 0 for m < i < n. Then the homotopy group ir X is determined as an abelian group by the boundary invariant /3 E H,1, m)(H X, Hm X, b+ 1X) of X since we have the isomorphism ir,, X =- ar{ /3t} in (6.4.6).

6.5 Split Eilenberg-Mac Lane functors

We describe Eilenberg-Mac Lane sequences. Such sequences are algebraic models of (m - 1)-connected (n + 1)-dimensional homotopy types X with iri(X) = 0 for m < i < n in the case when the Eilenberg-Mac Lane functors and H,(,m) are split.

(6.5.1) DefinitionLet n > m >_ 2. An Eilenberg-Mac Lane (m, n)-sequence S is an abelian group A together with a chain complex of abelian groups

S={H1 which is exact in H, 1(A, m) and H (A, m). Here we use the classical Eilenberg-Mac Lane functors H,(-,m) in Definition 6.1.2. A proper morphism between (m, n)-sequences is a homomorphism f: A-A' together with a commutative diagram

1W1 If. 1Ifs Hi

We call an (m, n)-sequence S free if H1 is free and injective if b is injective. 6A EILENBERG-MAC LANE FUNCTORS 185

Let S(m, n), resp. s(m, n) be the categories consisting of free, resp. injective, (m, n)-sequences and proper morphisms. These categories coincide with the categories S(EO, E,), resp. s(E0, E,) in Definition 3.6.1 where we set E0 =

H. ,(-, m). As in Definition 3.6.1 we have the natural k b equivalence relations and on these categories.

(6.5.2) Classification theoremLet 2 < m < n = m + r and assume or equivalently H,(,m) are split; this is the case for r < m or for m even and r < m; see Theorems 6.1.7 and 6.1.8. Then one has detecting functors

A: spaces(m, n) -> S(m, n)/

A': spaces(m, n )T - S(m, n) / A: types(m, n) - s(m, n)/

,1':types(m,n), -*s(m,n)/.

Proof We apply the classification theorem 3.6.3; compare the proof of Theorem 6.4.1 above.

As a special case of Theorem 6.5.2 one obtains for m >: 4, n = m + 2, the example discussed in Theorem 3.6.5. The theorem shows that (m, n)- sequences are algebraic models of homotopy types in case H,(,'°) or H(mj' are split. In fact, proper isomorphism classes of free (m, n)-sequences in S(m, n) arein1-1 correspondence (via A or A') with homotopy typesin spaces(m, n),. On the other hand, proper isomorphism classes of injective (m, n)-sequences in 9(m, n) are in 1-1 correspondence (via A or A') with homotopy types in types(m, n), Here we use the detecting functors in Definition 3.6.1(6), (7). Let X be the unique homotopy type in spaces(m, n) corresponding via A (or A') to a free (m, n)-sequence S as in Definition 6.5.1, so that A(X) = S (or A'(X) - S). In Definition 3.6.1(8) we associate with S the exact (- sequence which is the top row in the commutative diagram (A = H,,, X)

(6.5.3)

H1 -+ 0

II II at at II ,X -p [' X -p H,, X - 0.

This is a natural `weak' isomorphism of exact sequences. The same diagram is 186 6A EILENBERG-MAC LANE FUNCTORS available in case S is an injective (m, n)-sequence and X is a homotopy type in types(m, n),.

(6.5.4) Example For m = 4 and n = 7 we know that H(4) and H(8 are split and that H7(A,4) =A *Z/2 H8(A,4) = r(A) ®A ® Z/3; see (6.3.3) and (6.3.4). Hence 3-connected 8-dimensional homotopy types X with ir5X = 1r6X = 0 are in 1-1 correspondence with proper isomorphism classes of chain complexes in Ab S= (Hl which are exact in A * 7/2 and r(A) ®A ® 71/3 and for which H, is free abelian. If X is the homotopy type corresponding to S then one has the commutative diagram of exact sequences H, ->r(A)®A®Z/3-ker(S)cok(d)-*A*7/2-*0

II II at =1 II H8 X - r, X - ir, X -> H7X - r6 X -0 which is a natural `weak' isomorphism of exact sequences. For example, for A = 2/2 with r(71/2) = 71/4 the (4,7)-sequence S = {7171/4 -L 71/16171/2 -+ 0} corresponds to a 3-connected 8-dimensional homotopy type X with

2 * 4 0 71 71/4 71/8 71/8 71/2 , 0

II II II II II H8 X -> r, x - it7 X --* H7 X ---> r6 X -)'0 and ir6X= ir5X=0 and ir4X=H4X=71/2 and H6X=r5X=71/2, H5X= r4x=0.

6.6 A transformation from homotopy groups of Moore spaces to homology groups of Eilenberg-Mac Lane spaces

There is a connection between the homotopy groups of Moore spaces and the 6A EILENBERG-MAC LANE FUNCTORS 187

homology groups of Eilenberg-Mac Lane spaces. For example, it was observed by Eilenberg and Mac Lane, and J.H.C. Whitehead that

7r3M(A, 2) = FA = H4 K(A, 2).

We here describe further results of this kind. We have the canonical map k: M(A,n)->K(A,n) (n>2) which induces the identity on ir = A. This map induces a natural transformation Q which carries the homotopy groups of a Moore space to the homology groups of an Eilenberg-Mac Lane space (N > n):

7rNM(A,n) Q-b (CNk)i'-HN+1K(A,n)

= Ti = J,b (6.6.1) TNM(A, n) FNK(A, n). rN(k)

Here i and b are operators of the exact sequence of J.H.C. Whitehead which are isomorphisms for N> n. We call an element a E irNM(A, n) strictly decomposable if there is a homotopy commutative diagram SN - M(A,n)

(6.6.2)

XnN+-1 1 where X, i1 is a CW-complex which is n-connected and (N - 1)-dimensional.

(6.63) LemmaIf a E ITNM(A, n) is strictly decomposable then Qa = 0.

ProofIf a admits a factorization as above the composition

X11.11_1' ->M(A,n) -->K(A,n)N-1cK(A,n)N is null-homotopic since [X, ,1, K(A, n) = 0]. This implies the lemma.

(6.6.4) LemmaLet a E irNM(A, n) with N > n. Then Q[ a,,61 = 0 for any Whitehead product [a, 13 ].

Proof Let /3 E IrM M(A, n), M >- n. Then the composite

K(A,n)N+M-1 [a, p]: SN+M-1 __, SN V SM -M(A,n) - is trivial since n < N < N + M - 1, (n >- 2). 188 6A EILENBERG-MAC LANE FUNCTORS

For a, 13 EA = irnM(A, n), however, the element Q[a, 6] is not necessar- ily trivial. This follows from Theorem 6.6.6 below.

(6.6.5) TheoremFor N > 2n - 1 we have a commutative diagram:

TrN(Sn) xA -> ¶¶N(S n) ®A

IK.C2 ICI vrNM(A,N) --HN+1(A, n)

n TrN(M(A,n),M(A,n D j7TNM(A, n) -- cok c2.

Here j is defined by the n-skeleton M(A, n)n of M(A, n). Moreover, for a EA = 7rnM(A,n) and 6E ITNSn we define the function c1 by the composite c1Q, a) = a a 6. We show that Qc1 is bilinear, therefore we obtain the factorization c2. Moreover, we show that for the projection p above the composition pQ factors through j.

Proof of Theorem 6.6.5Qcl is bilinear. This follows from the left distributivity law and from Lemmas 6.6.4 and 6.6.3. Clearly by definition of j we have jc1 = 0. Moreover, by the Hilton-Milnor formula for irnM(A, n)n we see that the kernel of j is spanned by the image of c1 and by composites WW where W is a Whitehead product. Now we obtain the result by Lemmas 6.6.3 and 6.6.4 since N > 2n - 1.

For N = n + 2 the kernel of Q is actually generated by elements as in Lemma 6.6.4 and Theorem 6.6.5. The image of Q is a subfunctor which carries A to QirNM(A,n)cHN+1K(A,n). While the homology groups HJK(A, n) were extensively studied, the groups QIrNM(A, n) seem to be unknown. These groups are non-trivial since we get:

(6.6.6) TheoremFor n >- 2 we have the isomorphism

FA forn=2 Q: n+iM(A,n)-Hn+2K(A,n)=A®l2 forn>-3 and the surjection

I'T(A) for n = 2 Q: irrt+2M(A,n)-.Hn+3K(A,n)= A*Z2®A2(A)forn=3 A*772 forn-4. 6A EILENBERG-MAC LANE FUNCTORS 189

Proof The isomorphism is obtained since, by a result of J.H.C. Whitehead, r(HX) for any (n - 1)-connected space X. Below in Chapters 7, 8, and 11 we shall compute for any (n - 1)-connected space X. This yields the surjection, n >- 2.

The operator Q is available for F-groups with coefficients as follows (n

TrN(B, M(A, n)) HN+,(B, K(A, N)) Q-b 1k.i i 1= (6.6.7) lb FN(B, M(A, n)) FN(B,K(A,n)). k.

Here the isomorphisms i and b are given by the exact sequence in Section 2.3. The pseudo-homology is the group of homotopy classes of chain maps

(i) HN(B, K(A, n)) = [C* M(B, N),C* K(A,n)].

This is a bifunctor on pairs (B, A) of abelian groups. The operator Q is embedded in the following commutative diagram, the rows of which are short exact sequences:

(ii)

Ext(B,arN+,M(A,n)) >_+7rN(B,M(A,n)) µ Hom(B,ITNM(A,n)) IQ. IQ IQ. Ext(B,HN+2K(A,n)) ° HN+,(B,K(A,n)) Hom(B,HN+,K(A,n)).

Here Q* is induced by Q in(6.6.2). This diagram is easily obtained from the definition in (6.6.7), compare Section 2.3. For N = n + 1 the right column of (ii) is an isomorphism and the left column is surjective by Theorem 6.6.6. Therefore we get the

(6.6.8) CorollaryFor N = n + 1 diagram (ii) is a push-out diagram and

Q: 7rz+i(B,M(A,n))

is surjective,n >- 2. 190 6A EILENBERG-MAC LANE FUNCTORS

The fundamental importance of the subfunctors QIrNM(A,n) cHN+,K(A,n), QorN(B; M(A,n)) cHN+,(B; K(A,n)) is described by the following fact. Let X be any (n - 1)-connected CW- complex with7r,, X = H,, X = A. Then we have the homotopy commutative diagram M(A,n)-k -K(A,n)

(6.6.9)

where kX is the fundamental class of X and where i induces the identity on H,,. The map k is the one in (6.6.1). The map kX induces homomorphisms Q: FNX-*FNK(A,n)=HN+IK(A,n) Q: FN(B; X) -+ FN(B; K(A, n)) = HN+ l(B, K(A, n)) and (6.6.9) immediately implies:

(6.6.10) PropositionFor all (n - 1)-connected CW-complexes X with ir,, X = H X =A, n > 2, we have inclusions Q-rrNM(A, n) c QFNX c HN+,K(A, n),

i j

QTrN(B; M(A, n)) c QI'N(B; X) C HN+,(B; K(A, n)). i j

Here i is the identity if X = M(A, n) and j is the identity if X = K(A, n).

This fact clearly shows that a computation of I'NX involves the computation of QlrNM(A, n). For example we derive from (6.6.7) and Corollary 6.6.8.

(6.6.11) TheoremFor all (n - 1)-connected CW-complexes X with H,, X =A, n >- 2, we have surjective maps Q: rR+2X-"H,,+3K(A,n), Q: X) -. H,,, 2(B; K(A,n)).

In fact, we will compute the groups F,,+ 2X and F,,+,(B; X) for any (n - 1)-connected space X, n >: 2. 6B MOORE FUNCTORS 191 B MOORE FUNCTORS Moore functors are dual to Eilenberg-Mac Lane functors. We used Eilenberg-Mac Lane functors for the classification of (m - 1)-connected (n + 1)-dimensional homotopy types X which have trivial homotopy groups

7r,(X)=0for m

H,(X)=0 for m

We assume that m > 2 and n = m + r >- 4. For r = 2 such spaces are part of the classification in Chapters 8, 9, and 12. In both cases (*) and (* *) we use the classification theorem (Section 3.4). If we want to apply this theorem we have to choose a full subcategory C c typeset 1. In case (*) the category C is the category of Eilenberg-Mac Lane spaces K(A, m) which is equivalent to the category of abelian groups. In case (* *) the category C is the full homotopy category of `(m, n)-Moore types' which can be described in terms of the homotopy category M'" of Moore spaces M(A, m). There are algebraic categories equivalent to the category M"', for example for m >- 3 we have the equivalence G = M'" in Theorem 1.6.7. The classification theorem Section 3.4 describes bype and kype functors on C which incase (*) are the Eilenberg-Mac Lane functors and which in case (* *) are functors which we call 'Moore functors'. They are given by homotopy groups of Moore spaces. In the stable and metastable range we describe various algebraic properties of such Moore functors. For the metastable range we need the basic theory of quadratic functors which we describe in Sections 6.13 and 6.14.

6.7 Moore types and Moore functors Let n=m+r with m,r>2.

(6.7.1) Definition An (m - 1)-connected (n - 1)-type X is an (m, n)-Moore type if the homology groups of X satisfy Hi(X) = 0 for m < i < n. Let Moore(m, n) be the full homotopy category of (m, n)-Moore types in Top*/=.

In this section we describe an algebraic category equivalent to the category of (m, n)-Moore types and we study kype and bype functors on the category Moore(m, n) which we call Moore functors. Recall that Mm denotes the full homotopy category consisting of Moore spaces M(A,m) of degree m. Alge- braic models of this category are studied in Chapters 1 and 10. We now use 192 6B MOORE FUNCTORS homotopy groups 1M(A, m) to define an `enriched' category of Moore spaces which is equivalent to Moore(m, n).

DefinitionLet M(m, n) be the following category. An object is a pair (M(A, r n), i ), also denoted by (A, i ), where

is it (1) is a surjective homomorphism. For r >- 3 a morphism

(ilp, i/r): (M(A,m),i) -' (M(A',m),i) (2) is given by a homotopy class

;P E [M(A, m), M(A', m)] (3) (which is a morphism in Mm) and by a homomorphism r/r: 7r- 7r' such that the diagram `'.7rii-1M(A',m) 'nrt-1M(A,m)

1i li, (4)

41 7r 7r' commutes. For r = 2 a morphism is an equivalence class {ip, 41} of a pair (irp, ii) as above. Here we need the action + of Ext(A,7rm+1M(A', m)) on the set (3); see Section 1.3. Since for r = 2 we have m + 1 = n - 1 we thus get an action of Ext(A, ker i') on the set (3). Now we set (4p, 41) - if 1/i = 4io and if there is a E Ext(A, ker i') with irp = ipo + a. This equivalence relation defines the class (rlp, ip) which is a morphism in M(m, m + 1).

(6.7.3) PropositionFor n = m + r with m, r >- 2 there is an equivalence of categories 0: Moore(m, n) -- M(m, n).

This result is a consequence of proposition (111.8.8) of Baues [CH]. We obtain the equivalence in Proposition 6.7.3 as follows. For each Moore type X in Moore(m, n) choose a map

is M(A,m) -*X (1) which induces the identity A = H. X in homology. Then the functor 8 in Proposition 6.7.3 carries X to the pair

i . : 1M(A, m) -. 7r,,_ 1X) (2) 6B MOORE FUNCTORS 193 where i * is surjective since H- 1X = 0. Since r > 2 we can assume that i is the (n - 1)-skeleton of X. Then 0 carries a map f : X -> X' to (;p, 7rf) where fP is the restriction of f to the (n - 1)-skeleton. We now introduce functors on the category in Proposition 6.7.3 which we call 'Moore functors'.

(6.7.4) DefinitionThe Moore functors

Mo,M1:M(m,n)-+Ab

are defined as follows. For an object (A, i) = (M(A, m), i) let

M0(A,i) = kernel(i: 7ri_1M(A,m) -. rr)

and

M1(A,i) = cokernel('q: M0(A, i) - m)).

Here q is the restriction of the map q,*,: in_ 1M(A, m) - ir,, M(A, m) induced by the Hopf map n,,,. The function 77,*is a homomorphism since n = m + r >: 4. It is clear how to define Mo and M1 on morphisms.

We can describe the Moore functors also by use of homology groups. For this let X(A, i) be the (m, n)-Moore type corresponding to (M(A, m), i) via the equivalence 0 in Proposition 6.7.3.

(6.7.5) PropositionThere are isomorphisms

M0(A,i) M1(A, i) i) = i)

which are natural in (A, i) _ (M(A, m), i) E M(m, n).

Proof Since M(A, m) is the (n - 1)-skeleton of X(A, i) we have

rn-1X(A,i) _ -rri_1M(A,m).

Moreover the operator in-, in Whitehead's exact sequence coincides with i, that is in-1: ri_1X(A,i)= iriM(A,m) ir= 7ri_1X(A,i).

This implies H,, X(A, i) = ker(i) = MO(A, i) since 7r X(A, i) = 0. In a similar way we get X(A,i)-M1(A,i). 194 6B MOORE FUNCTORS

Next we define bifunctors

M#: M(m, n)°P x Ab -> Ab (6.7.6) M#: Ab°P X M(m,n) -> Ab, which we also call Moore functors. We define these functors by cohomology and pseudo-homology groups with coefficients: M#(A,i; B) =H"+'(X(A,i), B) M#(B; A,i) =H"(B,X(A,i)).

Here again we use the equivalence in Proposition 6.7.3. Hence M# and M# are kype and bype functors respectively with structure (M0, MI, 0, µ); see Proposition 6.7.5 and Theorem 3.3.9. Moreover M. and M# are dual to each other. We can describe the bype functor M# together with its bype structure (M0, M1, 0, µ) by the following push-out-pull-back diagram in which the rows are short exact.

(6.7.7)

Ext(B,lr"M(A,m)) H it"_i(B,M(A, m)) Hom(B,7r"_,M(A,m))

push II 19s 1 Ext(B, M1(A, i)) I'"_i(B,X(A,i)) - Hom(B,vr _1M(A,m))

II 1 Ext(B,M,(A,i))H M#(B;A,i) -Hom(B,M0(A,i)). Here q is the projection and j is the inclusion; see Definition. 6.7.4. One obtains this diagram similarly as in the proof of Proposition 6.7.5 by the exact sequence in Section 2.3. The diagram shows that M. can be computed by the homotopy groups of a Moore space with coefficients,

v,, _,(B, M(A, n)) = [M(B, n - 1), M(A, m)]. (1)

The top row in the diagram is the universal coefficient sequence. Though (1) is not a functor in B one can check that the push-pull group M#(B; A, i) in (6.7.7) is a functor in B, so that diagram (6.7.7) can be used to define the bifunctor M# in Proposition 6.7.6 in an alternative way. Diagram (6.7.7) yields a method of computation for M. Using duality the functor M# then determines the cohomology functor M# which in general by the definition in (6.7.6) cannot be easily computed.

As we mentioned already the category Moore(m, n) - M(m, n) can be 6B MOORE FUNCTORS 195

considered to be an algebraic category. For example we canuse the equivalence M' = G for m >- 3, see Theorem 1.6.7. Then one has to compute the functor ir,, _,: Mm = G -+ Ab in terms of G; this leads to an explicit description of M(m, n). We shall describe explicit examples below. For m >- 2 we compute the Moore functors on M(m, m + 2) in Chapters 8, 9, and 11.

6.8 On (m - 1)-connected (n + 1)-dimensional homotopy types X with H; X= 0 form < i< n

Let m, r >- 2 and n = m + r. The homology decomposition (Section 2.7) shows:

(6.8.1) LemmaLet X be an (m - 1)-connected (n + 1)-dimensional CW-space with Hi(X) = 0 form < i < n. Then there exists a map

f: V 1) ->M(HmX,m) such that X is homotopy equivalent to the mapping cone off.

We get the following application of the classification theorem (Section 3.4) for homotopy types as in the lemma. Let spaces(m, n)H be the full homotopy category of (m - 1)-connected (n + 1)-dimensional CW-spaces X with H, X = 0 for m < i < n. Moreover let types(m, n)H be the full homotopy category of (m - 1)-connected n-types Y with H,Y = 0 for m < i < n.

(6.8.2) Classification theoremLet n = m + r with m, r >- 2. Using the Moore functors M* and M* on M(m, n) in Section 6.7 one has detecting functors: A: spaces(m, n) H -+ Kypes(M(m, n), M*) A': spaces(m, n) H - Bypes(M(m, n), M*) A: types(m, n)H -+ kypes(M(m, n), M*) A': types(m, n) H -, bypes(M(m, n), M*).

Here the categories on the right-hand side are purely algebraic in case one is able to compute the Moore functors M* and M* respectively.

(6.83) Addendum For a space X in spaces(m, n)H the algebraicF- sequences associated with A(X) and A'(X) respectively are weakly isomorphic to the following part of Whitehead's exact sequence -F.X f,,-,X ,n-Iir - , X- 0

M,(A,i) 1ri_1M(A,m) `- IT 196 6B MOORE FUNCTORS where A = Hm X. The map i,, -I corresponds to the homomorphism i given by the (n - 1)-type P,_1(X) = X(A, i) of X; see Proposition 6.7.5. In particular it"X is determined by the object A(X) in Bypes(M(m, n), M#); see Remark 3.2.4.

Proof of Theorem 6.8.2 and Addendum 6.8.3. Consider Theorem 3.4.4 where we take C = Moore(m, n) = M(m, n). (*) Then we have spacesm 1(C) = spaces(m, n)H

types,;,(C) = types(m, n)H and E, resp. F, in Theorem 3.4.4 coincide with the Moore functors M#, resp. M# by use of the equivalence (*). Hence we get the theorem and the addendum by Theorem 3.4.4.

(6.8.4) Remark The classical Postnikov technique would use the detecting functor A for the classification of spaces X in types(m, n)H. Hence X is obtained as the fibre of a map X(A,i) - K(ir',n + 1) where X(A, i) = P-(X). This method, however, is not suitable for computation since it is very hard to compute the cohomology of the space X(A, i). The detecting functor A' in the theorem, however, needs only the Moore functor M. which is easier to understand and for which more methods of computation are available.

6.9 The stable case with trivial 2-torsion

We consider a special case of the classification theorem 6.8.2 for which the Moore functors are completely determined by stable homotopy groups of spheres. Let (6.9.1) Qk= lim1r"+kS"= rr"+k S"for k 2)

q A®n* i rj: A*Qr_Z®A®o-r_1 -'A®r-I -'A®Q-,CA ®A®or. 6B MOORE FUNCTORS 197

Here q is the projection and i is the inclusion and A * B denotes the torsion product. Using this notation we define the following category.

(6.9.2) Definition A stable r-sequence S is a chain complex in Ab of the form

b A*o,,_,®A®or, s H- p ------>R--_4 A*0r-2®A®a, q image(S) satisfying image(b) = kernel(a). Moreover all groups are finitely generated and A * 11/2 = 0 and cok(a) * 11/2 = 0. A morphism (cp, f, r) between such stabler-sequencesconsistsof homomorphisms(p E Hom(H, H'), f E Hom(A, A'), and r E Hom(R, R') compatible with b, a, and S. We obtain natural equivalence relationskandbon morphisms in the same way as in Definition 3.6.1. Let S(r) be the full category of stable r-sequences for which H is free and let s(r) be the full category of stable r-sequences for which b is injective.

We now give an explicit result which is a split case of the classification theorem 6.8.2; see Theorem 3.6.3.

(6.93) TheoremLet m > 3, r >_ 2, and n = m + r < 2m - 2. Then the homotopy types of (m - 1)-connected (n + 1)-dimensional finite CW-complexes X with H,(X) = 0 form < i < n and 11/2 * H* X = 0 are in 1-1 correspondence with isomorphism classes of stable r-sequences in S(r).

Let spaces(m, nYH be the full homotopy category of CW-complexes X as in the theorem and let types(m, nYH be the full homotopy category of n-types of such CW-complexes.

(6.9.4) TheoremLet m >: 3, r > 2, and n = m + r < 2 m - 2. Then one has detecting functors k A: spaces(m, n)H -p S(r)/

A': spaces(m, n)H --> S(r)/b

A: types(m, n)H -> s(r) /

A':types(m,n)H->s(r)/b .

Here the categories on the right-hand side are purely algebraicadditive categories, the sum being given by the direct sum of chain complexes.Also the categories on the left-hand side are additive categories since they are in the stable range. Moreover all functors in Theorem 6.9.4 are additive. 198 6B MOORE FUNCTORS

(6.9.5) Addendum Given a CW-complex X = Xs corresponding to the stable r-sequence S in Definition 6.9.2 there is a commutative diagram with exact rows, A = Hm X,

H,,X - F,_,X

II 11 11 I II 11 A*or ,®A®Q H -> --+ ker(6)-cok(d)->A*0,_20A0o_, -. cok(S). 17 image 6

The top row is Whitehead's exact sequence and the bottom row is deduced from the stable r-sequence S. Proof of Theorem 6.9.4 We consider the Moore functors M0, M,, M# on subcategories given by finitely generated abelian groups A, B with 7L/2 * A = Z12 * B = 0. Then we show in Theorem 6.12.15 below that the bype functor M. issplit and hence also the kype functor M# is split. Moreover by Theorem 6.12.15 one has the natural isomorphisms

m) =A * or-2 ®A ®or-, n) =A * or_, ®A ®or.

Hence the Moore functors M0, M, are given by

M0(A, i) = kernel(i: A * 0r-2®A ® (Tr_, -. ir) M,(A,i) = cokernel(ri: M0(A,i) -*A* or_, ®A (9 or).

Here r)is the restriction of rrin (6.9.1). Now one readily checks that S(r) a S(Mo, Ml) and S(r) - s(M0, M,) so that the results above follow from Section 3.6.

6.10 Moore spaces and Spanier-Whitehead duality The Spanier-Whitehead dual of a Moore space M(A,n) is again a Moore space in the case A is a finite abelian group. We here study the functorial properties of this duality between Moore spaces. Recall that Am m denotes the full homotopy category of all finite CW- complexes X = X," with dim X< n and trivial (m - 1)-skeleton X m In the stable range n < 2m - 1 Spanier-Whitehead duality is a contravariant isomorphism of additive categories

(6.10.1) D: A m -* A"m m. 6B MOORE FUNCTORS 199

This functor carries X to DX = X * and carries the homotopy class f E [ X, Y ] to Df =f * E [Y*, X*1 such that D: [X,Y] = [Y*, X*] is an isomorphism of groups for X, Y E AmThe isomorphism D satisfies DD = identity, that is X * * = X and f * * = f. The definition of D depends on the choice of (n +m)-duality maps DX: X* AX -> S"+m; compare Section 5.2. The dual of a sphere S°' is DS' = S" and the dual of the Moore space M(7L/t, m) of the cyclic group 7L/t is again such a Moore space,

(6.10.2) DM(7L/t, m) = M(7L/t, n - 1). For the pseudo-projective plane P, = S' U, e2 we have I."- 'P, = M(7L/r, n). This yields the function

I"-': [P, P,] [M(7L/r, m), M(Z/t, m)] between sets of homotopy classes. For cp E Hom(7L/r, Z/t) the element

(6.10.3) cp = B(q,) E [M(7L/r, m), M(7L/t, m)], m >- 3, is uniquely determined by the following two properties. First B(op) induces rp in homology and second B((p) lies in the image of the function I` above; see Theorem 1.4.4. We want to describe the dual of the map B(q'). For this we use the canonical identification

(6.10.4) 7L/t = Ext(7L/t,7L) which yields the isomorphism Hom(7L/r,7L/t) _ Hom(7L/t,Z/r) carrying cp up to p * = Ext(gp, 7L).

(6.10.5) PropositionThe duality map DX with X = M(7L/t, m) and t > 1 can be chosen such that the isomorphism D: [M(7L/r, m), M(7L/t, m)] _ [M(7L/t, n - 1), M(7L/t, n - 1)] carries B(q) to B(qp*). Proof Given the duality map DX for allt > 1 one gets a derivation S: FCyC--* Ext(-, ®1/2) by setting with i and q as in (1) below

S(rp)i71"- iq One can check by Theorem 1.4.8 that S: Hom(Z/r,7L/t) -+ Ext(l/t,7L/r ® 7L/2) is a homomorphism and that 5(ip)=8(ep*). This shows that the 200 6B MOORE FUNCTORS derivation S is completely determined by the values 6(X,) E 1/2 where X,: 7L/2'+ 1 .7L/2' is the projection, r >_ 1. We now alter the duality map DX with X=M(7L/2',m) by the element er= where q: X* AX- Sn+m+1 is the pinchmap for the top cell. Then we see that these new duality maps D' yield a derivation S' with S'(X,)=s(X,)+X*er+l -(Xi)*e,=0.

In fact, we have S( X,) = (X,) * er by definition of er and X,*--,,, = 0. Hence also S' = 0 and the proposition is proved. For the Moore space M(7L/t, m) = Sm U, em+1 we have the inclusion i and the pinch map q such that

Sm >+M(7L/t m) (1) is a cofibre sequence. The Spanier-Whitehead dual of the inclusion is the pinch map and the dual of the pinch map is the inclusion. Hence the cofibre sequence

S"+7 M(7L/t, n- 1)aS"-1 (2) is the dual of the sequence (1) above. This shows that D in (6.10.4) satisfies

(3) where 17m E lrm Sm is the Hopf map. This formula determines the isomor- 1 phism D (Proposition 6.10.5) completely. We now consider more generally Moore spaces of abelian groups A. Recall that for an abelian group A we have the group

(6.10.6) G(A) _ [M(7L/2, n), M(A,n)],n >- 3 together with the extension

A ® 71/2°G(A) A *7L/2 (1) given by A *7L/2 CA ->A ® Z/2; see Definition 1.6.6. The extension is used for the definition of the category G; objects in G are abelian groups and morphisms A -* B are pairs (V, fir) with gyp: A -+ B E Ab and iJr: G(A) --> G(B) E Ab such that A((p ® 7L/2) = /iO and (gyp * Z 12)A = p4,. For the abelian group G(A, B) of such pairs (V, r/i ): A -- B we have by Theorem 1.6.7 the isomorphism [ M(A, n), M(B, n)] = G(A, B), (2) 6B MOORE FUNCTORS 201 which in fact is given by an equivalence of categories Mn a G, n >_ 3. We define the group

(6.10.7) G(A) = [M(A, n), M(7/2, n)] which plays a similar role to that of G(A) in (6.10.6).

(6.10.8) PropositionThere is a natural isomorphism

G(A) = Hom(G(A),71/4) for which the following diagram commutes. Ext(A,71/2)r--°>G(A) µ-» Hom(A,Z/2)

II II II Hom(A *7/2,7/4) N Hom(G(A),l/4) µ» Hom(A (9 7/2,7/4).

In the bottom row we set A = Hom( p., 7/4) and µ = Hom(O, 7/4).

Proof By (6.10.6X2) we have [M(A, n), M(7/2, n)] = G(A,7/2) __ Hom(G(A), Z/4) where the right-hand isomorphism carries ((p, i) to +G. In fact, 0 determines cp by the composite

gyp: A ->A ®7/2A G(A) -L7/4 which clearly maps to the subgroup Z/2 of 7/4. Compare also Lemma 8.2.7.

We use the isomorphism in Proposition 6.10.8 as an identification.

(6.10.9) PropositionFor a finite abelian group A and A* = Ext(A, 7) there is a duality isomorphism of abelian groups ,r: G(A*) = Hom(G(A).7/4) for which the following diagram commutes. AOG(A*) A* ®7/2) µ *A* *7/2

II II 111 µ Ext(A,7/2) -% Hom(G(A),Z/4) -- Hom(A,7/2). 202 6B MOORE FUNCTORS

The left- and right-hand side are the canonical isomorphisms; see also (6.11.4) below. Proof We have by (6.10.1) and (6.10.7) the sequence of isomorphisms G(A#) _ [M(Z/2,n),M(A*,n)]

T D Hom(G(A),Z/4) _ [M(A,m - 1),M(Z/2,m - 1)]. Since D is compatible with 0 and µ by Proposition 6.10.5(2) we get the commutative diagram in the statements of the proposition. Let fAb c Ab be the full subcategory of finite abelian groups. Then (6.10.10) #: fAb -- 14 fAb, A --A# = Ext(A,71) is a contravariant isomorphism of categories with (A#)# =A. On the other hand, we have the subcategory f M" c M" of Moore spaces of M(A, n) with finite A. For this category we obtain by (6.10.1) the duality isomorphism (6.10.11) D: 1M"' = 1Mn-' which carries M(A, m) to M(A#, n - 1); see (5.2.11). The isomorphism of categories G = M', n >t 3, thus yields the composite D: fG =1M" fMn-' =fG where fG c G is the full category given by finite abelian groups A. This duality functor for fG is computed in the next result. (6.10.12) TheoremThe duality functor D: fG - fG carriesA to A* and carries (gyp, 40: A --> B to the morphism (cp#, T-' Hom(+i, Z/4)T): B# -A* where T is given by Proposition 6.10.9. Proof Let F: M(A, m) --> M(B, m) correspond to ((p, 41) and let (cp *, q * ) correspond to F*: M(B#, n - 1) --> M(A#, n - 1). Then we compute 4*: G(B#) --), G(A*) as follows. Let 8G(B#) = [M(Z/2, n - 1), M(B#, n - I.A. Then i * (/3) = F* -,8. Now F *P = (13 *F )* with /3 * = T(8) yields the result since $ *F = Hom(#, Z/4X T (/3 )).

6.11 Homotopy groups of Moore spaces in the stable range

We consider the homotopy groups and cohomotopy groups of Moore spaces it"M(A,m) = [S", M(A,m)], (6.11.1) irmM(B,n - 1) = [M(B,n - 1),Sm] 6B MOORE FUNCTORS 203

in the stable range m < n < 2m - 1.If A isfiniteabelian and B = Ext(A,7L) =A# these homotopy groups are Spanier-Whitehead dual to each other, since we have the isomorphism D: [S",M(A,m)] = [M(A#,n - 1),Sm] (1) by Theorem 6.4.1. More generally we get the isomorphism D: [X, M(A,m)] = [M(A#,n - 1), X*] (2) for X E Am m. Here the right-hand side is a homotopy group with coefficients in A# which we described as a functor in Theorem 1.6.4. Hence we are able to obtain the dual result for the functor M(A, n) - [X, M(A, n)]. We first obtain the following dual of the universal coefficient sequence.

(6.11.2) PropositionLet X be a CW-complex with dim X< 2m - 1 and let A be an abelian group. Moreover suppose that X is finite or that A is finitely generated. Then there is the binatural short exact sequence:

A® [X,Sm] [X,M(A,m)] -µ31A*[X,Sm+1 ]

Here A * B = Tora(A, B) is the torsion product of abelian groups.

d Proof Let C N D -A be a short exact sequence where C and D are free abelian. Then the cofibre sequence M(C,n) d M(D,n)->M(A,n)-*M(C,n+1)-+ (1) induces for dim X < 2n - 1 the exact sequence C®[X,S"]dotD®[X,S"]->[X,M(A,n)] _+ C®[X,S"+1]doI D®[X,S"+'] This sequence is obtained by applying the functor [ X, -I to the cofibre sequence (1). For this we use the fact that we have a natural isomorphism [ X, M(C, n)] = C ®[ X, S" ] (2) in case C is free abelian. Here we need the assumption that X is finite or that A, and hence C, is finitely generated. If A is a finite abelian group and X E AR -" we get by (6.11.1X2) the commutative diagram of `coefficient sequences':

.1 9 A#®zr"(X) '-1 [X,M(A#,n)] -3A#,,"+1(X) (6.11.3) sto aID zIo Ext(A,TrmX*) H 7rm_,(A,X*)--w Hom(A,TrnX*). 204 6B MOORE FUNCTORS

Here the left-hand side and the right-hand side are given by the duality isomorphisms 7 r nX - 7 rmX *, 7r n + 1(X) = 1rm_ 1X * and by the binatural isomorphisms l A# ®B = Ext(A, B), (6.11.4) A#*B=Hom(A,B) where A is finite, A, B E Ab.

Proof of (6.11.3)For a mapping cone Cf we obtain the dual DCf = Cg by a map g respresenting f *. We can apply this to the mapping cone of d in Proposition 6.11.2(1). This now yields (6.11.3) since pinch map and inclusion behave as in Proposition 6.10.5(1), (2).

Let FM' be the subcategory of Mm consisting of Moore spaces M(A, m) with A finitely generated. Then FG is the corresponding subcategory of G. We consider the functor

(6.11.5) 17X: FG = FM' -> Ab which carries A to [X, M(A, m)]. Here we assume that X is m-connected and dim X < 2m - 1. The functor is the analogue of the functor 7rn(-, X) in (1.6.9). We can compute the functor 7TX up to natural isomorphism in terms of the homomorphism 71: 1rm+1(X)®71/2-->7rm(X) (1) induced by the Hopf map qm E 7rm + (S'), that is 71(a ® 1) _ qm a for a E 7rm+1(X)=[X,Sm+11.

(6.11.6) Definition We define for A E FAb the abelian group G(71, A) with 71: 7r ® 71/2 -> 7r' by the push-out diagram: Ext(7r*, A (9 71/2) N G(7r*, A) ---> Hom(7r*, A)

II A®7r®7 l/2 push

1Ion

A®7r' G(71, A) µ > A * 7r. Here we assume that 7ris a finite abelian group so that we get for 7r* = Ext(7r,71) the binatural identification in the diagram by (6.11.4). More- over G(7r *,A) is the group of morphisms 7r * --*A in the category G. The diagram is natural in A E FG and hence yields a functor G(71, -): FG -* Ab. The next result shows that this functor is naturally isomorphic to 7r x in (6.11.5). 6B MOORE FUNCTORS 205

(6.11.7) TheoremLet X be a finite CW-complex with dim X< 2m - 1, m >: 3, and let rl: 7Tm+'(X) ®71/2 -> 7rm(X) with rl(a (g1) _ima be given by the Hopf map 7Im. Moreover assume 7rm+'X is finite. Then one has for a finitely generated abelian group A an isomorphism [X,M(A,m)]=G(q,A) which is natural in A E FG and for which the following diagram commutes.

A ®7rm(X) [X, M(A,m)] A * irm+I (X)

II -1 µ II A®lrm(X) G(q,A) .A*,rm+1(X).

The isomorphism is not natural in X. We point out that7r m +'(X) in Definition 6.11.6 is automatically finite if X is (m + 1)-connected. We can apply Theorem 6.11.7 in particular for the case that X = S" is a sphere. Thus we can compute it"M(A,m) for n < 2m - 1 in terms of r,*: Ir"Sm+' only.

Proof of (6.11.7)Since we assume 7r"'+'X to be finite the group G(-q, A) is well defined. If A is finite then the theorem is the Spanier-Whitehead dual of the result in Theorem 1.6.11. In fact in this case we have the commutative diagram

Ext(7r*, A (9 11/2) N G( ,A) -> Hom(Ir*, A)

II 11 A®7r(& Z1/2 D A*ir

II 11 Ext(A*,7r(9 71/2) N G(A*, ir) -. Hom(A*, ir) where D is the duality isomorphism of Theorem 6.10.12. This diagram is natural in A E 1G. Now we can apply an argument as in the proof of Theorem 1.6.11. This proves the proposition for finite A; moreover it is easy to extend the isomorphism to the case of finitely generated A.

(6.11.8) CorollaryLet X be a finite CW-complex with dim X< 2m - 1, m >- 3, and 7rI+'X finite. Then the extension in Theorem 6.11.7 is split if and only if one of the following three conditions is satisfied: (a) A has no direct summand Z/2; (b) 7T' "X has no direct summand 71/2; (c) each element a E 7r'"+'X, generating a direct summand Z/2, satisfies elm a = 2 a' for some a'. 206 6B MOORE FUNCTORS

Hence, if (a), (b), or (c) hold, one has an isomorphism of abelian groups (unnaturally)

[X, M(A, m)] =A * vrin+ 1(X) ®A ®am(X ).

ProofIf (a) or (b) is satisfied the top row in the diagram of Definition 6.11.6 is split; if (c) holds the bottom row in Definition 6.11.6 is still split, hence the corollary is a consequence of Theorem 6.11.7.

6.12 Stable and principal maps between Moore spaces

We describe some properties of the stable homotopy groups (m < n < 2m - 1)

(6.12.1) 1r,m)(A,B)=-rr(A,n,B,m)=[M(A,n),M(B,m)].

We assume that A and B are finitely generated abelian groups. Using the universal coefficient sequences one has the following commutative diagram in which rows and columns are exact sequences and in which

7r n+1sm+1 1T= lrnsm =

m (6.12.2)

7r-= 1rn-1sm = rnsm+ are stable homotopy groups of spheres. (6.12.3) 0 0

e Ext(A,B®a+) - B®irmM(A,n) Hom(A, B ®ir) - 0

0 -Ext(A,irn+,M(B,m)) -Lar(A,n,B,m) µ Hom(A,ar,,M(B,m))-"0

lµ. 1" lµ 0 Ext(A,B*a) --B*orm+'M(A,n)-----*Hom(A,B*lr_)

1 1 0 0 The column in the middle is obtained as a special case of Proposition 6.11.2. 6B MOORE FUNCTORS 207

Since A and B are finitely generated abelian groups we have natural isomorphisms: Ext(A,B®ir,)=B(& Ext(A,Tr,) (1) Hom(A,B(9 Ir)=B®Hom(A,7r) (2) (6.12.4) EWA, B * ar) =B *Ext(A,zr) (3)

Hom(A, B * Tr_) = B * Hom(A, ;r_). (4)

Proof of (6.12.4)It is clear how to obtain these isomorphisms in the case that all groups involved are cyclic groups. In order to prove naturality we give the following definition of the isomorphisms in (6.12.4). Let A, - A0 -* A and B, >- B0 -. B be short free resolutions of A and B respectively, where A0, A, B0, B, are finitely generated. For (2) let

41: B ® Hom(A, ir) --> Hom(A, B ®Tr) be defined by $b 00 _'Pb with cpb(a) = b ® rp(a). We obtain (1) be the commutative diagram = Hom(A,,B®7r,)B®Hom(A'Tr,)

Ext(A,B(& a,) - B®Ext(A,a,).

Next we get (3) by the commutative diagram

B, ®Ext(A , Tr) = Ext (A , B, (9 7T)

U I B *Ext(A, ir).-- EWA, B * ar).

Similarly we get (4) by the commutative diagram = Hom(A, B, (9 7r_) B, ®Hom(A , a_ )

i U Hom(A,B*a_) --. B*Hom(A,ir_).

We use the isomorphisms in (6.12.4) as identifications. This way we get the

(6.12.5) Lemma Diagram (6.12.3) commutes. In fact, the top row and the bottom row are induced bythe(A, µ)-extension for irmM(A, n) and 208 6B MOORE FUNCTORS

7rn'+ 1M(A, n) respectively. The left-hand column and the right-hand column are induced by the (A, µ)-extension for 1rn+M(B, m) and irn M(B, m) respectively. 1 Moreover diagram (6.12.3) is natural in M(A, n) E Mn and M(B, m) E Mm respectively.

(6.12.6) DefinitionLet 7r, 7r' be abelian groups and let71:7r --> 7r' be a homomorphism with 71(27x) = 0. We associate with a homomorphism T(7)): Hom(A, B * 7r) -* Ext(A, B (9 7T') which is binatural for A, B E Ab. Let T(71) be the composite Hom(A, B * 7r) -* B * Hom(A, 7T)

IT(17) la Ext(A, B (& 7r') F- B ®Ext(A, Ir'). Here the horizontal arrows are defined as in the proof of (6.12.4); they are isomorphisms if A and B are finitely generated. Moreover d is the boundary in the six-term exact sequence of homological algebra induced by the extension Ext(A, 7r') >- G(A, 71) -. Hom(A, 7r ) in Definition 1.6.10. Now consider again diagram (6.12.3). We want to describe the kernel of A * in Ext(A, B (9 7r+) and the image of A,, in Hom(A, B * 7r_ ). The Hopf maps induce homomorphisms, see (6.12.2),

71_: 7r_-> 7r, 71_= 71n-1 = (7)m)* 71+ 7r- 7t++7)+= In = (11m)* which satisfy the condition on 71 in Definition 6.12.6 so that T(71_) and T(71+) are defined. Now the exact sequences in diagram (6.12.3) show by Definition 6.11.6 and Theorem 1.6.11 respectively: (6.12.7) Lemma The kernel of A * in Ext(A, B (& 7r+) is the image of T67+) and the image of µ * in Hom(A, B * 7r_) is the kernel of T(7) _ ). In the lemma we again assume that A and B are finitely generated. The operator r(71) has a natural interpretation as a Toda bracket. To this end we recall the classical definition of Toda brackets as follows.

(6.12.8) DefinitionLet

/ II W Y>X-iY-'Zy 6B MOORE FUNCTORS 209 be a diagram in Top* where H: 0 and G: a/3 = 0 are homotopies. Then the map

TH.G:'W- Z is defined by the addition of homotopies, that is TH G = - aH + G(I x y) where I = [0, 1] is the unit interval. The Toda bracket (a, /3, y) is the subset of [YEW, Z] consisting of all TH.G with H: a/3 = 0 and G: /3y = 0. If the group [1W, Z] is abelian the Toda bracket (a,/3, y) is a coset of the subgroup

G = a* [XW,Y ] + (Yy)*[Y',X, Z] C [I.W, Z] or equivalently (a, /3, y) is an element of the quotient group [LW, Z]/G. The element (a, /3, y) depends only on the homotopy classes of a, /3, and y. The subgroup G is called the `indeterminancy' of the Toda bracket (a, /3, y ).

We consider the following example which is associated with stable maps between Moore spaces.

(6.12.9) ExampleLet A and B be abelian groups and let

Al>- A0 -. A, d, (1) B,) >Bo-.B be short free resolutions. For an abelian group it we have the exact sequence

B*7r>- (2) where the homomorphism in the middle is dB ® Tr. We now consider maps between Moore spaces (m < n < 2m + 1) d, M(A1,n) -M(A0,n) a M(B,,m) MOO, m) where a is an element

aEHom(A0,B, (9 7r)=[M(A0,n),M(B,,m)] (3) with 1r = lrSm. We have adA = 0 iff dA a = 0 in Hom(A,, B, ®a) and we have dBa - 0 if (dB ® 7r)* a = 0 in Hom(A0, B0 0 77-). This shows that the Toda bracket (dA, a, dB) is defined if and only if there is an element a E Hom(A, B * -rr) with

a=iaq: a--4 B*ir>B,®ir. (4) 210 6B MOORE FUNCTORS

Moreover (dA, iaq, dB) is an element in the group

Ext(A, B ®rr') = [M(A1, n + 1), M(Bom)]/G (5) with 7r' = ii+ 1Sm. Here the indeterminancy G is the sum of the images of (dB (&1r') * : Hom(A1, B1 ®ir') -- Hom(A1, Bo ®vr')and (dA)*: Hom(A0, B0 (9 ir') --> Hom(A1, B0 (9 ?r') so that the definition of Ext yields equation (5). Hence the Toda bracket -r(a) _ (dA, iaq, dB) defines the function T:Hom(A,B*ir)-*Ext(A,B®ir') (6) which is a homomorphism and natural in A, B E Ab.

(6.12.10) PropositionLet

-a = q,: Tr= TrTSm -> 7r' = ?r.Sm be induced by the Hopf map rt with m < n < 2m - 1. Then the Toda bracket r in Example 6.12.9 coincides with the natural transformation T(77) in Definition 6.12.6. Here we restrict T and T(17) to finitely generated abelain groups.

The proposition yields a further interpretation of the operators T(,q_) and T(rj+) in Lemma 6.12.7. We shall not make use of the result in Proposition 6.12.10 so that we can omit here its somewhat elaborate proof. We do not know whether the restriction to finitely generated abelian groups is necessary. Next we want to study `principal maps' between Moore spaces. To this end we recall the definition of mapping cone and principal map; see Baues [AH]. For a pointed space U let CU be the reduced cone of U, that is CU is the quotient space CU = I x U/(I x * U {1} x U). The mapping cone Cg of a map g: U V in Top* is defined by the push-out diagram CU - Cg

with io(x) _ (0, x) for x E U.

(6.12.11) DefinitionConsider a diagram x U I jg (1) Y `-a V 6B MOORE FUNCTORS 211 in the category Top* of pointed spaces where H: of = gu is a homotopy. Then the triple (u, v, H) yields the map between mapping cones, C(u, V, H): Cf --> Cg. (2) This map carries y E Y to igv(y) and carries (t, x) E CX, 0< t:5 Z, to iH(2t,x).g Moreover (2) carries (t, x) E CX withz 5 t < 1 to irg(2t - 1, u(x)). We call this map and any map homotopic to (2) a principal map. Let PRIN(f, g) c [C1,Cg] (3) be the subset of all homotopy classes represented by principal maps. The properties of principal maps are studied in Baues [AH].

Again let AI) -d" )A0 -.A and BIB>B0 -*B be short free resolutions of A and B respectively. Then the Moore spaces M(A, n) and M(B, m) are mapping cones of the maps dA: M(A1,n)- M(A0,n) andde: respectively. Hence we get by Definition 6.12.11 the subset (6.12.12) PRIN(dA, dB) c [M(A, n), M(B, m)] of principal maps between Moore spaces. This subset is a subgroup which is natural for maps a: M(A, n) -* M(A', n) and b: M(B, m) - M(B', m) since such maps are always principal and since the composition of principal maps is principal. We consider dA as a chain complex concentrated in degree 0 and 1. Similary dB ®ir:B,®Ir-+B0®Ir is a chain complex concentrated in degree 0 and 1. Hence we have the abelian group of homotopy classes of chain maps [dA, dB ® 1r] which in an obvious way yields a bifunctor in A, B E Ab.

(6.12.13) Lemma For all abelian groups A, B, IT one has an isomorphism of abelian groups [dA,dB ®Ir]=Ext(A,B*Ir)ED Hom(A,B®a). This isomorphism is natural in A, B, IT provided A or B are finitely generated or IT is a field. Proof The classification of chain maps yields the natural short exact sequence Ext(A,B*ar) N [dA,dB®-rrI -*Hom(A,B(9 1r) 212 6B MOORE FUNCTORS which is split as a sequence of abelian groups. For finitely generated A or B we obtain a natural splitting by the commutative diagram (see (6.12.4))

[ dA, -rr ] ®[7L, dB ] = Hom(A, ir) ® B

is =1 [dA,dB ® ,r] ->Hom(A,B(9ir).

Here it and Z denote chain complexes concentrated in degree 0. The function s carries {} ® {a} to the homotopy class of the composite of chain maps

A, 0 B,® -rr

I I I Ao- £ 7r=7L®7r -Bo®7r. a®a

If zris a field we consider the following diagram where 7L[B] is the free abelian group generated by B. Moreover dB is obtained by B, N Bo = Z[B] Pw B.

K(A, B, ir) c [dA,7L[B] ® -rr] = Hom(A,7L[B] ® ar) eI push I(P®ar). I Ext(A,B*ir) C[dA,dB®7r] -Hom(A,B(&ir).

Here K(A, B, ir) = kernel(p (9 1r)* is a functor in A, B, ir. Since IT is a field p 0 -rris split surjective and therefore the subdiagram push is a push-out diagram. One can now use naturality of 0 to show that 0 is trivial. Hence by naturality of the diagram we obtain a natural splitting which, if A or B are finitely generated, coincides with s above.

(6.12.14) TheoremLet A and B be finitely generated abelian groups and let 71_ and q+ be given as in Lemma 6.12.7 with m < n < 2 m - 1. Then one has the following two exact sequences which are natural in A and B E G.

PRIN(dA,dB)>-' rr(A,n,B,m)-Hom(A,B*a_)T( Ext(A,B®7r) Hom(A,B*zr) T(ni)-Ext(A,B®irt)LPRIN(dA,dB) A-0[dA,dB ®ir]. 6B MOORE FUNCTORS 213 Here µ = µ* p. and 0 = A * 0 are given by the operators in (6.12.3). Moreover A carries the principal map C(u, v, H) to the chain map d,, -* dB IT given by u and v. The homotopy class of this chain map is well defined by the homotopy class of C(u, v, H). The results in chapter V of Baues [AH] yield a proof of Theorem 6.12.14.

(6.12.15) TheoremLet m < n < 2m - 2 and let A and B be direct sums of cyclic groups with A *71/2 = 0 and B *71/2 = 0. Then one has an isomorphism 0: [M(A,n),M(B,m)]Ext(A,B(9 7r+)ED Ext(A,B*7r) ®Hom(A,B(9 7r)®Hom(A,B*7r_) which is natural in A and B. Here 7r+ = 7r"+, S'", 7r = 7r" S'", and 7r_ = 7r"_ ,S'" are stable homotopy groups of spheres as in (6.12.2). Equivalently all rows and columns in (6.12.3) are short exact and naturally split provided A and B satisfy the assumptions above. As usual we write A = (71/k)a if A is a cyclic group of order k with generator a EA. An element a (-= IT in a finite abelian group IT generates the subgroup (71/1 a I) a where I a Iis the order of a. A subset E c IT is a basis if I a Iis a prime power for a e E and

(D (l/1 a1)a- 7r aEE is an isomorphism. For the proof of Theorem 6.12.15 we introduce the following elements.

(6.12.16) NotationLet E, c o,,be a basis of the stable group Q, = 7r"S'n = m + r < 2n - 1. Moreover let a c- Er be an element of orderI a I = p`("' where p is a prime. We choose for a stable maps --(a), (a ), 77(a), and p(a) between elementary Moore spaces with the properties below, where A = and where i is the inclusion and q is the pinch map. eo(a)=a:S"-Sm 88(a) =ia: S" -S' - M(A,m) (1) e°(a) = aq:M(A, n -1) - S" -> S°' e(a) =iaq: M(A,n - 1) -+S" -VS' - M(A,m). We choose a map 6'(a): S"+' - M(A, m) with q,,°(a) = la and let 6(a) _ C°(a)q: M(A,n) -S"+' - M(A,m). (2) 214 6B MOORE FUNCTORS

Let 770(a): M(A, n) --> S' with i *i0(a) = a be the Spanier-Whitehead dual of 6°(a) and let

n(a)=iii0(a):M(A,n)->S'-M(A,m). (3) Moreover we denote by p(a) a map

p(a): M(A,n+1)-*M(A,m) (4) satisfying i * p(a) = t; °(a) and q,, p(a) = a ,q0(a ). The element p(a) exists if and only if for A = B = (7L/p`(°))a c Q, we have A=Hom(A,A* A) cimage iicHom(A,A* see Theorem 6.12.14. In particular,ifI a l * 2 the element p(a) exists. Moreover for I a I * 2 we choose the elements (1)-(4) to be elements of order I a 1; forI a I = 2 we choose such elements with minimal order. Now let B, B' be Z or cyclic groups of prime power order and let X E Hom(B, A) and x' E Hom(A, B') be the canonical generators which yield maps between Moore spaces, also denoted by X and X' respectively, see Theorem 1.4.4.

(6.12.17) LemmaAppropriate compositions of X, X' and elements (1)-(4) above generate the abelian group [M(B, d), M(B', d')] ford < 2d' - 2.

(6.12.18) LemmaI f a lis odd the element p(a) is Spanier-Whitehead self-dual, that is Dp(a) = p(a ). Proof Since f °(a) is the dual of r70(a) we see that p(a) - Dp(a) is in the image of 2i = AA*in Theorem 6.12.14, that is, there is a E Ext(A, A ® it+) with A = Z/I a I such that p(a) - Dp(a) = 0(a). This implies DA(a) =D(p(a) -Dp(a)) =Dp(a) - p(a) On the other hand, 0(5) = i aq:

M(A, n + 1) 1 Srt+2 --5-1. S"` M(A, m) is self-fual, that is D(iaq) = iaq, so that we get 27100 = 0. If Ial is odd this implies a = 0.

Proof of Theorem 6.12.15 We have a basis E,+, c it+, E, C IT, resp. E,_, C Tr_in the groups of the theorem. Then there is a unique natural isomorphism 0 in the theorem which carries (for A, B (=- (7L,Z/p`(°)} and a (-= E,+ E E,_,) the elements chosen in (6.12.16) to the corresponding basis elements of the right-hand side in 6.12.15 given by a. Hence the isomorphism 0 is determined by the choice of elements in (6.12.16). 6B MOORE FUNCTORS 215 6.13 Quadratic 71-modules

We here introduce the `quadratic algebra' which is needed for the metastable range of homotopy theory; for a more extensive treatment see Baues [QF]. A ringoid R is a category for which all morphism sets are abelian groups and for which composition is bilinear (a ringoid is also called a 'pre-additive category' or an `Ab-category'). A biproduct in a ringoid is a diagram (see Mac Lane [C])

r_ (6.13.1) X 1XvY r, Yj, which satisfies rl i, = 1, r2'2 = 1, and i, r, + i, r, = 1. Both sums and products in a ringoid are canonically equipped with the structure of a biproduct. An additive category is a ringoid in which biproducts exist. Clearly the category Ab of abelian groups is an additive category with biproducts given by direct sums A ® B of abelian groups. A functor F: B -> S between ringoids is additive if (6.13.2) F(f +g) = F(f) + F(g) for morphisms f, g E R(X, Y). Moreover we say that F is quadratic if the function A defined by (6.13.3) A(f,g)=F(f+g)-F(f)-F(g) is bilinear. Itis clear that an additive functor carries a biproduct to a biproduct. Let Add(7L) be the full subcategory in Ab consisting of finitely generated free abelian groups. Additive functors F: Add(7L) --> Ab are in 1-1 correspondence with abelian groups; the correspondence is given by F - F(7L). In fact one readily obtains the following equivalence of categories.

(6.13.4) Lemma The category of additive functors Add(Z) - Ab with natural transformations as morphisms is equivalent to the category Ab. The equivalence carries F to F(7L) and the inverse of the equivalence carries an abelian group A to the functor ®A: Add(7L) - Ab, B - B ®A, given by the tensor product of abelian groups.

We now introduce `quadratic Z-modules' which are in 1-1 correspondence with quadratic functors Add(7L) - Ab. In this sense a quadratic 7L-module is just the `quadratic analogue' of an abelian group. Moreover quadratic Z- modules allow precisely the quadratic generalization of Lemma (6.13.4) above; see Theorem 6.13.12.

(6.13.5) Definition A quadratic 7L-module M = (Me H > Me P + Me) 216 6B MOORE FUNCTORS is a pair of abelian groups Me, MeC together with homomorphisms H, P which satisfy the equations PHP = 2 P and HPH = 2 H. A morphism f: M -* N between quadratic 7L-modules is a pair of homomorphisms f.: Me - Ne, fCe: MCe -f Nee which commute with H and P respectively. Let QM(77) be the category of quadratic 71-modules. This is an abelian category.

For a quadratic 7L-module M we define the involution

(6.13.6) T=HP-1: M1e-'Mee Then the equations for H and P in Definition 6.13.5 are equivalent to PT=P and TH=H. We get TT=1 since 1+T=HP=HPT=T+T2. We define for n c- Z the function

n*: ML->MM (6.13.7) n * (x) = nx + (n(n - 1)/2)PH(x),x E Me.

One can check that (n m). = n * m * and (n + m),, = n,, + m * + nmPH. Let i/n = i/ni by the cyclic group of order n >: 1. We call M a quadratic (i/n)-module if n MeC = 0 and n * Me = 0. Let QM(i/n) c QM(i) be the full subcategory of quadratic (71/n)-modules. We identify a quadratic 7L-module M satisfying Me = 0 with the abelian group Me. This yields the full inclusion of categories

(6.13.8) Ab c QM(Z) which carries an abelian group A to the quadratic 7L-module (A --> 0 -A) which we also denote by A. Moreover we have the following canonical functors on QM(i).

(6.13.9) DefinitionThere is a duality functor D: Qm(Z) - QM(i) with D(M) given by the interchange of the roles of H and P respectively, that is

D(M) = ((DM), 2"_110 (DM),, P' (DM),) (1) with (DM), = MeC, (DM)ee = M, H° = P, and P° = H. Clearly DD(M) = M. Moreover an additive functor A: Ab --* Ab induces a functor A: QM(i) - QM(i). Here we define A(M)11=A(Mfe) and A(M), =A(ML) with H and P given by A(H) and A(P) respectively. For example the functor ® C: Ab -> Ab, C E Ab, carries M to H®lMee®C P®1*Me®C). M®C=(ML®C ) (2) 6B MOORE FUNCTORS 217

This tensor product should not be confused with the quadratic tensor product C ® M = C ®Z M defined in Definition 6.13.13 below.

The following construction yields many examples of quadratic 7L-modules.

(6.13.10) DefinitionLet F: R -' Ab be a quadratic functor and let X V Y be a biproduct in R. The quadratic cross-effect F(X I Y) is defined by the image group, see (6.13.3),

F(X I Y) = image(A(i1r,,'2 r2): F(X V Y) --> F(X V Y)}. (1)

If R is an additive category this yields the biadditive functor

F(I) ): R X R -* Ab. (2)

Moreover we have the isomorphism

( D ( D (3) which is given by F(il), F(i2) and the inclusion i12: F(X I Y) c F(X V Y). Let r12 be the retraction of i12 obtained by r/r-' and the projection to F(X I Y). For the biproduct X V X one has maps µ ='I + i 2: X -i X V X and V = r1 + r2: X V X --- X. They yield homomorphism H and P with

F{X} = (F(X)'" F(X I X) -F(X)) (4) by H=r12F(µ) and P=F(V)i12. Nowwe derive from f+g=V(fVg)µ the formula F(f+g)=F(f)+F(g)+PF(flg)H (5) or equivalently A(f, g) = PF(f I g)H.

(6.13.11) PropositionLet F: R -> Ab be a quadratic functor where R is an additive category. Then F{X}, X (=- R, is a well-defined quadratic 7L-module and X - F{X} defines a functor R - OM(7L).

Proof We define the interchange map t=i2r1 +i1r2: XvX->XVX.

Then tµ = µ and Vt = V. Moreover t induces an isomorphism

T: with F(t)i12= i12T andr12F(t) = Tr12. Hence we get TH = H and PT = P. 218 6B MOORE FUNCTORS

Moreover we obtain HP = 1 + T be applying F to the commutative diagram in R X v X -°-> x µ X v X 1µvµ Ivvv

XvXvXvX IvTvl xvxvxvx. Here we use the biadditivity of F(l).

The significance of quadratic 7L-modules is now described by the following quadratic generalization of Lemma 6.13.4; see Baues [QF].

(6.13.12) TheoremThe category of quadratic functors Add(7L) - Ab with natural transformations as morphisms is equivalent to the category QM(7L) of quadratic 7L-modules. The equivalence carries F to F{71) and the inverse of the equivalence carries the quadratic 7L-module M to the functor ®Z M: Add(7L) - Ab, A -A ® M, given by the quadratic tensor product below. Hence we have a 1-1 correspondence between quadratic functors F: Add(7L) -> Ab and quadratic 7L-modules. In particular any quadratic functor F: Add(7L) --> Ab is completely determined (up to isomorphism) by the fairly simple algebraic data of a quadratic 7L-module M = F(7L). Theorem 6.13.12 is one of the reasons to study the following quadratic tensor product and the corresponding quadratic Hom functor.

(6.13.13) DefinitionLet A be an abelian group and let M be a quadratic 7L-module. Then the quadratic tensor product A ®d M is the abelian group generated by the symbols a ® m, [a, b] ® n with a, b E A, m E M, n E M«. The relations are: (a+b)®m=a®m+b®m+ [a, b]®H(m); [a, a]®n=a®P(n); a ® m is linear in m; [a, b] ® n is linear in a, b and n respectively. These relations imply [a, b] ®n = [b, a] ®T(n) (*) where T = HP - 1 is the involution on M«. The quadratic tensor product is a functor ®Z : Ab x QM(i) - Ab. Induced functions f®g:A®ZM-A'®ZM' 6B MOORE FUNCTORS 219 are defined by (f®g)(a(&m)=(fa)®(gem) and (f®gX[a,b]®n)= [fa, fb] (9 (g,,n). We point out that the quadratic tensor product is compatible with direct limits in Ab and QM(71) respectively. We also write A ® M = A (&z M.

Proof of (*) [b, a] ®T(n) = [b, a] ®HP(n) - [b, a] ®n =(b+a)®P(n)-b®P(n)-a®P(n)-[b, a] ®n =[b+a,b+a]®n-[b,b]®n- [a,a]®n-[b,a]®n

= [a, b] ®n. 11 (6.13.14) DefinitionAgain let A be an abelian group and let M be a quadratic 7L-module. A quadratic form a = (ae, aL,): A -> M is a pair of functions ae: aee:AxA -'Mee with the following properties (a, b C= A): ae(a + b) = ae(a) + ae(b) +PaCe(a, b) aee(a,a) =Hae(a) a,C is 77-bilinear. These properties imply a,,(a, b) = Taee(b, a) (*) where T = HP - 1 is the involution on Mee. Let Hom1(A, M) be the set of all quadratic forms A -i M. This is an abelian group by (a,a,,)+(j3 Hence we obtain the quadratic Horn functor Hom1: Ab°P x QM(77) -* Ab. Induced functions are given by the formula Hom(f, g X a) = /3 with fe = g, ae f and Pee-gee aee(fxf). Proof of (*) Taee(b, a) = HPaee(b, a) - aC1(b, a) =H(ae(b+a) - ae(b) - ae(a)) - aeC(b,a)

= aef(b + a, b + a) - aeC(b,b) - aee(a, a) - aee(b, a)

= aCe(a, b). 220 6B MOORE FUNCTORS

Restricted to the subcategory Ab c OM(71) the quadratic tensor product and the quadratic Hom functor coincides with the classical (linear) tensor product and Hom functor respectively. The next lemma is well known in the linear case.

(6.13.15) LemmaLet C be a finitely generated free abelian group. Then one has for *C = Hom(C, Z) the isomorphism

X: (*C) ®z M= Homg(C, M) which is natural in C E Add(71) and M E OM(71).

Proof We define X as follows. For a, b E *C let X(a (9 m) = a = (ae, aec) be given as follows (x, y E C) ae(x) = a(x)m + (a(x)(a(x) -1)/2)PH(m), aPe(x, y) = a(x)a(y)H(m).

Moreover let X([a, bJ ® n) = 6 = (Ne, /a) be defined by 13e(x) =a(x)b(x)P(n), /3Ce(x, y) =a(x)b(y)n +a(y)b(x)Tn.

The lemma shows the next result which is an addendum to Theorem 6.13.12.

(6.13.16) PropositionFor each quadratic functor F: Add(7L) --> Ab one has a canonical natural isomorphism F(C) = C ® M, C E Add(71).

Here M = F(71) is a quadratic 7L-module. For each quadratic contravariant functor F: Add(7L) - Ab one has a canonical natural isomorphism F(C) = Hom5(C, M), CE Add(Z), with M = F*(71). Here F* is the covariant functor defined by F*(C) = F(*C) with *C = Hom(C, Z).

The ring 71 of integers in the discussion above can be replaced by any ring R or even by a small ringoid R; this is done in Baues [QF]. For example, we get for the ring R = 71/n the following result corresponding to Theorem 6.13.12. Let Add(7L/n) be the full subcategory of Ab consisting of finitely generated free 71/n-modules. 6B MOORE FUNCTORS 221

(6.13.17) TheoremThe category of quadratic functors Add(77/n) - Ab with natural transformation as morphisms is equivalent to the category OM(771n) of quadratic i/n-modules. The equivalence carries F to F{7L/n} and the inverse of the equivalence carries the quadratic Z/n-module M to the functor 0 M: Add(Z/n) -b Ab, A -A 0 M given by the quadratic tensor product.

We observe that the quadratic tensor product A ®z M and the quadratic Homz(A, M) are both additive in M and quadratic in A. The quadratic cross-effects are given as follows. We obtain the inclusion

(6.13.18) A®B®Mee=(AI B) ®z M="' >(A(D B)®z M by it2(a ®b ®m) = [ila, i2b] ®m. Moreover we get the induced maps

A®B®MeC-LB®A®M,,, (1)

A®0M A®A®MeeP>A®1M. (2) Using the involution T = HP - 1 on MeC they are defined by H(a®m)=(a®a)®H(m), H([a,b]®n)=(a®b)®n+(b®a)®T(n), T(a®b(9 n)=b®a®T(n) P(a®b®n)=[a,b]®n. Clearly (2)isthe quadratic 7L-module {A} ®z M defined in Definition 6.13.10(4). On the other hand, we have the projection (6.13.19) Hom(A®B,Mef)=Homz(AIB,M) Homz(A®B,M) which carries a = (ae, a,,) to /3: A ®B - Mee with 6(a ® b) = aef(i,a, i,b). Now the structure maps for the cross-effect (6.13.19) are the homomorphisms Hom(A ® B, Mee) T-4Hom(B ®A, M,,), (1) H- Homz(A, M) Hom(A ®A, MCe) Homz(A, M). (2) Again using the involution T = HP - 1 on Met they are defined by (T$)(a ®b) = T(/3(b ®a)) (Ha)(a 0 b) = a,(a, b) + Taee(b, a) (PI3)e(a) = H13(a ®a) (P/3),(a,b) = f3(a (9 b). 222 6B MOORE FUNCTORS

Here (2) is the quadratic 7L-module Hom7({A}, M) defined in Definition 6.13.10(4). Any quadratic functor F: R - Ab satisfies by Definition 6.13.10(3) the formula

(6.13.20) F(XI V V Xr) = ® F(X;) ® ®F(X; I X.). i «j Using this formula we get by (6.13.18) and (6.13.19) similar formulas for (A)ED ®Ar)®eMand Homg(AtED (D Ar,M).Since Z®dM=Me and Homp(7L, M) = Me we in particular get the following isomorphisms of abelian groups where Tr" = 7r ® ... ® Tr denotes the n-fold direct sum.

(7L") ®z M = (M') - ® (MCe)"(" -1)/2= Hom(7L", M). Not every quadratic functor F: Ab - Ab is of the form A -A ® M. We always have, however, the canonical natural transformation (6.13.21) A: A ®P F{7L} - F(A), A E Ab, defined as follows. For a E A leta: Z -A be the homomorphism with a(1) = a. Then we getfor m E F(7L) and n E F(7L 17L)the formulas A(a (9 m) = F(aX m) and A([a, b] (9 n) = PF(a I LX n). By (6.13.20) and (6.13.16) the map A is an isomorphism if A is a finitely generated free abelian group. We call A the tensor approximation of the quadratic functor F: Ab - Ab. Similarly we obtain a Hom-approximation of any quadratic functor G: Ab°P - Ab. This is a natural transformation (6.13.22) A: G(A) -),Hom,(A,G(7L}), A E Ab, which is an isomorphism if A is finitely generated and free. Here G{7L} is defined for R = Ab°P as in Definition 6.13.10; equivalently we have G{7L} _ G*{7L} with G* as in Proposition 6.13.16. We now consider the important classical quadratic functors

(6.13.23) which appear frequently in the literature. Here ®2is the tensor square defined by the tensor product in Ab ®2(A) =A ®A. (1) The functor S2 is the symmetric square given by S2 (A) =A ®A/{a (9b- b ®a -- 0). (2) Moreover A2 is the exterior square AF (A)=A®A/{a®a-0} (3) and (i2(A) =A 0 A/{a 0 b +b 0 a - 0}. (4) 6B MOORE FUNCTORS 223

Next P2 is the quadratic construction defined by the quotient

P2(A) = 0(A)/A3(A) (5)

where.(A) is the augmentation ideal in the group ring 71[ A] and A3(A) its third power. We can define Whitehead's F functor as a quotient

F(A) = P2(A)/{y(a) - y(-a) - 0) (6) where y: A --> P2(A) carries a to the element represented by a- 1 E A(A). The composite y: A - P2(A) -» F(A) coincides with the universal quadratic map y in (1.2.1). We say that a function f: A - B between abelian groups is weak quadratic if

[a,b]f=f(a+b)-f(a)-f(b) (7)

is bilinear for a, b E A. Moreover f is quadratic if in addition f(-a) = f(a) for a, b E A. The function y is universal weak quadratic; that is, each weak quadratic function f: A - B admits a unique factorization f = f °y where f ° : P2(A) --> B is a homomorphism. On the other hand y is universal quadratic, that is, each quadratic function f: A --> B' admits a unique factorization f =f °y where f ° : F(A) - B is a homomorphism. Each quadratic functor F: Ab - Ab determines the quadratic 7L-module F{7L} by Definition 6.13.10(4). For the functors F in (6.13.22) we in particular get the following isomorphisms of quadratic 71-modules:

®2{71} 7L®(71 (8) p2(Z)

Here 7LP = D71® is the dual of 7L®; see Definition 6.13.9. Moreover we get

S-{Z) = ZS = (Z -' 7L 7L) (9) F{71} a Zr = (7L I Z 2 7L) which are again dual quadratic 7L-modules. Next we obtain

A2{71} 71^ =(0-*Z->0) (10) id(Z) Z). Here id is the identity functor of Ab and Z is the quadratic 7L-module given by the inclusion (6.13.8). Hence 7L is the dual of V. Finally we get

®2{7L} -P(1)= (71/2-° Z - 71/2). (11) 224 6B MOORE FUNCTORS

The following result gives a new interpretation of the classical functors above.

(6.13.24) TheoremFor each functor in (6.13.22) the tensor product approximation is an isomorphism. Hence for A E Ab one has natural isomorphisms

®2(A) =A ® l®, S2(A) =A ® is, A2(A) =A (2) _7A,

®2(A) =A ®P(1), P2(A) =A ®71P, T(A) =A ®71r.

The torsion functor F: Ab --> Ab with F(A) =A * A, however, is a functor for which the tensor approximation is not an isomorphism, in fact F(Z) = 0 in this case. The theorem above raises the question of classifying all indecomposable quadratic 11-modules. For this recall that an object X in an additive category is indecomposable if X admits no isomorphism X = A ® B with A * 0 and B * 0. It is an interesting problem of representation theory to classify all indecomposable quadratric 71-modules which are finitely generated as abelian groups. There is the following result where we say that a quadratic 71-module M is of cyclic type if Me and M« are cyclic groups.

(6.13.25) PropositionThe quadratic 11 modules below together with their duals furnish a complete list of indecomposable quadratic 71-modules of cyclic type. Let s, t -- 1 and let C = 71 or C = 71/p` where p = prime, i >_ 1.

H P « Me Me M C 0 C 2 C C C 1 2'- 1/2' 0 1 2'- 0 71/2' Z/2' 11/2' s+t> 1 2'- 2- 71/2' Z/2' Z/2' s+t> 1 /2s+ t 2 2s+ I 11 Z/2' Z/ 2s+ 1 2'-'+2'-'+1 2 2$+ l/ Z/2' Z/ s>1 6B MOORE FUNCTORS 225

With respect to the two tensor products of Definitions 6.13.13 and 6.13.9(2) we have the following rule.

(6.13.26) PropositionFor a quadratic Z-module M and for abelian groups A, B we have the natural isomorphism A®Z(M®B)=(A(81M)®B.

Here ®dis the quadratic tensor product and M 0 B is defined in Definition 6.13.9(2). Moreover ® on the right-hand side denotes the usual tensor product of abelian groups.

6.14 Quadratic derived functors

In this section we associate with a quadratic functor F a quadratic chain functor F*. Thedefinition of F* is motivated by properties of homotopy groups of Moore spaces which we exploit in the following section (6.15). The chain functor F. is used here for the definition of derived functors. The F-chain functor F,, in Definition 6.2.5 is a special case of F* . We obtain two `quadratic torsion products', A *'M and A*" M, which are derived from the quadratic tensor product in Section 6.13. For a further discussion of such quadratic derived functors we refer the reader to Baues [QF]. Let R be a ringoid with a zero object denoted by 0. A chain complex X * = (X*, d) in R is a sequence of maps in R d... (6.14.1) Xn-1 (nEi) with dd = 0. A chain map F: X* -+ Y* is given by maps F = F,,: X -a Y with dF = Fd and a chain homotopy a: F = G is given by maps a = a,,: Xi_ 1- + Y, with -F + G = a,, d + d ai + 1. LetChain(R) be the category of chain complexes in R and let Chaln(R)/ = be its homotopy category. A chain complex X * is concentrated in degree n, n + 1, ... , m with n< m if X, = 0 for i < n and i > m. We also need the category Pair(R) of pairs in R; objects are morphisms dA: Al -*A0 in R and morphisms are pairs F = (F1, FO) for which the diagram

(6.14.2)

commutes. Hence Pair(R) is a full subcategory of Chain(R) consistingof chain complexes concentrated in degree 0 and 1. A homotopy a: F = G of 226 6B MOORE FUNCTORS maps F, G: dA -> de in Pair(R) is a map a: A0 --> B, with -F, + G, = adA and -Fo+Go=dB a.

(6.143) DefinitionLet R be an additive category and let F: R - Ab be a quadratic functor. Then we define the induced quadratic chain functor

F* : Pair(R) --> Chain, = Chain(Ab). (1) For an object dA: A, - A0 in Pair(R) let F*(dA) be the following chain complex of abelian grops concentrated in degree 0, 1, 2

F(A, IA,) -- F(A,) ®F(A, IAo)aF(A0)

F2(dA) F,(dA) Fo(dA).

The boundary maps d,, d2 are given by P in the quadratic Z-modules F{A1) and F{Ao} respectively, see Definition 6.13.10(4), namely

d, = (F(dA), PF(dA IA0)), (2)

d2 = (P, -F(AI I dA)). (3)

One readily checks d, d2 = 0. In Baues [QF] (6.4) we show:

(6.14.4) TheoremThe quadratic chain functor F. in Definition 6.14.3 induces a functor F* : Pair(R)/= -* Chain1/= between homotopy categories.

We now consider the case R = Ab. Using short free resolutions we obtain the full and faithful functor

(6.14.5) is Ab -> Pair(Ab)/= as follows. We choose for each abelian group a short exact sequence Al - A0 -.A where A0 and A, are free abelian. For a homomorphism gyp: A - B we choose a map F,,: dA --> dB in Pair(Ab) which induces (p. Then the functor i carries A to dA and carries cp to the homotopy class {F,) which depends only on gyp. Using (6.14.5) and Theorem 6.14.4 we are now ready to define derived functors of a quadratic functor Ab - Ab. 6B MOORE FUNCTORS 227

(6.14.6) DefinitionLet F: Ab --> Ab be a quadratic functor. Then one obtains the derived functors L,F:Ab--' Ab (t=0,1,2) as follows. We define L,F by the homology group (L,FX A) = H,(F* dA) where dA = iA is chosen as in (6.14.5). That is, L, F is the composite of the functors

H Ab Pair(Ab)/= F'-> Chain1/= Ab.

The remarks (6.5), (7.5) in Baues [QF] show that L,F coincides with the derived functor considered by Dold and Puppe [HN].

We now obtain derived functors of the quadratic tensor product as follows.

(6.14.7) DefinitionLet A be an abelian group and let M be a quadratic 7L-module. Then we have the quadratic functor

®M:Ab-+Ab which carries A to the quadratic tensor product ((&M) (A) =A ®Z M. Hence the derived functors L,(®M) are defined by Definition 6.14.6 above. We call

A *"M= (L2(®M))A (1)

and A *'M = (L,(®M))A (2)

the quadratic torsion products. Using (6.14.11) below we get ®M = LO(OM), that is

A ®M= (Lo(®M))A (3)

is the quadratic tensor product. The quadratic tensor products are obtained more explicitly by the following definition where dA: A, -A, is ashort free resolution of A. Consider the chain complex (0 M) * dA:

A, ®A, ®M«-az A, ®Z M ®A, ®Ao ®M« .`''.> A0 ®Z M (4)

which is defined by the boundary operators

d2=(P,-A, ®dA®M«) andd, _(dA®z M,P(dA(9A0®M«)). 228 6B MOORE FUNCTORS

Here P is given as in (6.13.18X2). Then we have

A ®M = cokernel(d,) (5) A *'M = kernel(d,)/image(d2) (6)

A *" M = kernel 02)- (7) All functors in (5), (6), (7) are additive in M and quadratic in A. The quadratic cross-effects are:

(AIB)®M=A®B®Mee (1)

(6.14.8) (A I B)*'M=H,(dA (9 dBIMe,) (2) (A IB)*'M=A* B * MeC. (3) Here dA ® dB is the tensor product of chain complexes. The Kiinneth formula yields a natural exact sequence

(A*B)®MCe> H,(dA®dB,Mee)-"(A(9 B)*Mee (4) which is split (unnaturally). There is a natural isomorphism

H, WA ® dB, Mee) = Trip(A, B, MeC) (5) where the right-hand side is the triple torsion product of Mac Lane [Ti']. Moreover one has a natural injective homomorphism (6.14.9) A*"MNA*A*Mee. The results in Remark 6.2.8 and Theorem 6.2.9 are proved in 7.7 and 7.8 of Baues [QF]. We shall need the following `right exactness' of the quadratic tensor product. Let M, - Mo -* M -* 0 be an exact sequence of quadratic Z- modules in QM(71) and let A be an abelian group. Then the induced sequence

(6.14.10) A®IM, '0 i A®ZM,, "'A®aM,0 is exact. Here, however, 1 ® i need not be injective in case iis injective. Moreover let A,dA0-A - 0 be an exact sequence of abelian groups. Then the induced sequence (6.14.11) A,®M®A,0A00Me1_____*Ao®Mg® A®M-0 is exact where d, = (d ® M, P(d (D A0 (D MCe)). For example for M = 7Lr we obtain the exact sequence in Lemma 1.2.8 by (6.13.24). 6B MOORE FUNCTORS 229 We point out that for any quadratic functor F: Ab -p Ab the derived functors L, F, t >- 1, depend only on the restriction of F to the subcategory of free abelian groups. If F commutes with direct limits we have F(A°) =A° 0 M for any free abelian group A0. Here M = F{7L} is the quadratic Z-module in Definition 6.13.10(4). Hence we obtain in this case (L,F)(A) =A *'M (6.14.12) (L2F)(A) =A *"M so that the quadratic torsion products suffice to describe the derived functors of Dold and Puppe. The equations in (6.14.8) hold for any quadratic functor F if A is finitely generated.

6.15 Metastable homotopy groups of Moore spaces

We consider homotopy groups ?rmM(A, n) of a Moore space M(A, n), n >- 2. In the stable range m < 2n - 1 these groups are computable in terms of stable homotopy groups of spheres. We are here mainly interested in the metastable range m < 3n - 1. In general it is an unsolved problem to describe the groups ormM(A, n) only in terms of properties of homotopy groups of spheres. In addition one has the problem of describing these groups as a functor on the category M' of Moore spaces in degree n. For the stable range we describe partial solutions in Section 6.6; moreover for m = n + 2, we obtain complete solutions in Chapters 8, 9, and 11. More generally we shall deal with the homotopy groups (6.15.1) ir,, M(A, n) = [F,mK, M(A, n)1 where EmK is the m-fold suspension of a CW-complex K and m >: 2 so that (6.15.1) is an abelian group. Clearly for K=S° this is the homotopy group Trm M(A, n). The group (6.15.1) yields the functor

Trm : Mn --> Ab (1) where the homotopy category M', n >- 3, of Moore spaces is an additive category with the sum given by M(A,n)VM(B,n)=M(A(D B,n). (2) The left distributivity law of homotopy theory shows that the functor ;rm is additive for dim(lmK) < 2n - 1. Moreover the functor 1rm is quadratic for dim(Y.mK) < 3n - 2. We now consider the quadratic cross-effect of this functor. To this end we need for CW-complexes X, Y the Whitehead product map w:IXAY-- 1XV Y (3) 230 6B MOORE FUNCTORS where X A Y = X X Y/X V Y is the smash product. This map induces on homotopy sets the operation

[XX, U] X [MY, U] -> [I(X n Y),U] (4) which carries (a, /3) to the Whitehead product [a, /3 ] = w*(a,,6). For the inclusions i 1: 1X c I X V I Y and i 2: l Y c I X v MY we thus have 1'11'21= w. We define the space

M(A I B, n) = IM(A, n - 1) A M(B, n - 1). (5) Since M(A, n) = MM(A, n - 1) we thus have, as a special case of (3), the Whitehead product map

[i1,i2J: M(A I B, n) -->M(A,n) V M(B,n). (6)

Now the Hilton-Milnor theorem shows that in the metastable case the functor 1r,,has the following cross-effect.

(6.15.2) Lemma For dim(Y.mK) < 3n - 1 there is a binatural isomorphism

1rm M(A I B, n) = 1rm (M(A, n) I M(B, n)) which carries a to the composite [i1, i, ]a. Using this isomorphism the quadratic 7L-module an {M(A, n)} in Definition 6.13.10(4) coincides with

1T,nM(A,n) H'1T'M(AIB,n)

Here H = y2 is the Hopf invariant and P = [1, 1] * is induced by the Whitehead product [1, 11 where 1is the identity of M(A, n), that is P(a) = [1,1]a. Moreover T = - (I T,1) * is induced by the interchange map T21.

This lemma yields many interesting examples of quadratic 7L-modules. In particular we get for spheres the quadratic 71-modules (m < 3n - 2)

(6.15.3) lrm{Sn} = (1TmS" H 7rmS2n-1 P irSn)

Here H is the classical Hopf invariant for homotopy groups of spheres and P = [ in, tn ] *is induced by the Whitehead square[ t,,, t,, ] of the identity t E 1rSn. The operators H and P in (6.15.3) are known in many cases. For example we get by inspection of Toda's book [CM] the following list. Let be the direct sum in QM(77), that is

M®N=(M,®N,H®N,Me,®N« P®PM,®N,) 6B MOORE FUNCTORS 231 and recall that an abelian group A yields the quadratic 2-module A= (A - 0->A).

(6.15.4) List of am{Sn}:

Ts2n-1 P sn (n,m) TrmsnH (2,3) yr (3,5) 71^ ®71/2 (3,6) (7L/4 -L 71/2-0 1/4)s 7/3 (4,7) (71®71/4-' 0712' 7®71/4)®7/3 (4,8) Z P (9 71/2 (4,9) 71P®71/2

(5,9) (71/2 -°-)- 71 71/2) (5,10) 71s 71/2 (5,11) 71^ ® 71/2 (D Z/2 (5,12) (71/4--2/8-°71/2) ®71^ ®71/3 ®Z/3 ®Z/5 (6,11) is (6,12) 71A(& 71/2®71/2

(6,13) (71/4 -° 71/2 71/4)6) 71/3 ® 71/5 (6,14) Zr ®71/8 ®7Ls ®Z/3 ®71/2 (6,15) 2/2 ® Z/2 ® Z/2

Similarly we get as in (6.15.4) the quadratic 71-module

7rm VmKS2n-1 P 1rmKSn) K(Sn) = (1rKSnm for dim(ImK) < 3n- 1;see Lemma 6.15.2. Using the quadratic tensor product in (6.13.13) one has the following crucial result.

(6.15.5) TheoremLet A be a free abelian group and let dim(EmK) < 3n - 1 with m, n 2- 2. Then there is an isomorphism

irn M(A, n) =A ®Z irn (Sn) which is natural in A and K.

Proof Since both sides are compatible with direct limits itis enough to consider finitely generated free abelian groups A. For these the proposition is a special case of Proposition 6.13.16. 0 232 6B MOORE FUNCTORS

We can extend the isomorphism in Theorem 6.15.5 in an appropriate way to all abelian groups. For this we define a functor I',, as follows.

(6.15.6) DefinitionLet K be a finite dimensional CW-complex and m, n >- 2. Then we obtain a functor which carries an abelian group A to the abelian group F, (A, n). If A is free abelian we have f,K (A, n) = irnM(A, n). If A is not free abelian we choose a d short free resolution A 1'-' A0 -.A which yields the cofibre sequence M(AI,n) - _ M(A0,n) -M(A,n) where M(A, n) is the mapping cone of d = dA. Consider the maps (d, 1) (0.t).,7rnM(A0,n) 7rnM(A16) A0,n) where 1 is the identity of M(A0, n) and 0 is the rivial map. Then we get the quotient group r,n(A, n) = ir, M(A0, 01(d, 1)* kernel(0, 1) which defines the functor above. As in (6.14.5) let F, = (F1, F0): dA - dB be a pair map which induces c: A --> B. Then F, induces the homomorphism cp * : r,, (A, n) - F,n(B, n) which carries the coset of 6 E irnM(A0, n) to the coset of Fol:.

(6.15.7) Lemma The induced map cp * is well defined.

Proof We have to check that cp * does not depend on the choice of F". If F; = (F;, Fo) is also a map which induces (p we have a homotopy a: F,o = F,o and hence Fo -- F0 + dB a. Now the left distributivity law of homotopy theory (see Section A.9 in the appendix) yields the formula E c"(F0,dBa)y"f n22 which shows that F, 'C - F0 C is an element of (dB,1)* kernel(0,1)..

The next result is a corollary of Theorem 6.15.5.

(6.15.8) Corollary LetA be an abelian group and let dim(Y.mK) < 3n - 1 with m, n > 2. Then there is an isomorphism F,n (A, n) =A ®z irn($") which is natural in A and K. 6B MOORE FUNCTORS 233

The corollary indicates that it should be possible to generalize the quadratic tensor product in such a way that an isomorphism as in (6.13.7) is also availableif dim(TmK)3n - 1. Then, however, r, (A, n) need not be quadratic in A.

Proof of (15.8) Since 7r"k,is quadratic we have the commutative diagram

(d, 1), 7r,n M(A0, n) ® 7r, M(A1 IA0, n) = ker(0,1),, ) 7r,nM(A0, n)

II 9, A0 ® 7rn{S"} ®A1 ®A0 1 A0 ® 7rnK {S").

Here dl coincides with dl in (6.14.11). Hence quadratic right exactness in (6.14.11) yields the result.

(6.15.9) NotationLet k z 0 and n >_ 2. We obtain as a special case the functor F,,': Ab - Ab.

Here we set rk(A) = rR (A, n) where K = Sk is the k-sphere. Hence (6.15.8) yields for k < 2n - 1 the natural isomorphism

F (A) =A ®7r"+k(S").

Now the list in (6.15.4) makes it easy to identify these functors. For example we get for m = 11, n = 5, k = 6 the natural isomorphism F56(A)=A®(71A®Z/2®Z/2) =A2(A)®71/2®A®71/2 where A2(A) =A ® 71^ is the exterior square; see Proposition 6.13.26. For k = 0 we get F,°(A) =A and for k = 1 we have

I r(A) for n = 2 F,1 (A) =A®71/2forn_3.

This is Whitehead's functor. Whitehead proved that an (n - 1)-connected

CW-space X satisfies r"+ 1X = r (H" X), n >_ 2. We compute in Chapter 11 the functor F2 which is not quadratic. This is needed for the computation of the homotopy group ir4 M(A, 2); see Theorem 11.1.9.

The relevance of the functor F, in Definition 6.15.6 arises by the following result. 234 6B MOORE FUNCTORS

(6.15.10) TheoremLet K be finite dimensional and let n, m > 2. Then there is a homomorphism A: F',' (A, n) - ir, M(A, n) which is natural in M(A, n) and K and which maps surjectively to the image of the map i: ir,, M(A0, n) - iTm M(A, n) induced by the inclusion i in Definition 6.15.6.

Proof Let X1 = M(A1, n) and X0 = M(Ao, n) such that M(A, n) is given by the push-out diagram

C X 1-- M(A,n) U U1 X1 e X0 where CX1 is the cone of X1. This yields the following commutative diagram with exact rows a ir,K 1(CX1 v Xo, X1 V Xu) im(X1vXo)2

I(Vd,1). I(d,1).

a vrRK+1 M(A n) -2-1 ir,R (M(A,n),X.) ,.M(A,n)

1 a F.K, (A, n). Here we set ir,, (X1 VXo)2 =kernel{(0,1) irn (X1 vXa) - zrn(X0)}. The boundary a in the homotopy exact sequence of a pair yields the isomorphism in the top row. Hence (d, 1). carries kernel (0, 1). to the image of d which is the kernel of i. Therefore the factorization A of i exists. Using the diagram in the proof we define the functor

I (6.15.11) 1'T,,,(A, n) = (Trd,1)*a-' kernel(d,1), One has the inclusion FT,R (A,n) cjirm 1M(A,n) (1) which is natural in M(A, n). Using Theorem 6.15.10 we see that

F',R+1(A,n) A 7r,+IM(A,n)ijiTn 1M(A,n)-0 (2) 6B MOORE FUNCTORS 235

is exact, so that jirm IM(A,n) can be identified with the cokernel of A in Theorem 6.15.10.

(6.15.12) Lemma The functor (6.15.11) is well defined.

Proof Let ip: M(A, n) -> M(B, n) be a realization ofgyp: A - B. Let y E Trn IM(A, n) with jy E n). Then there exists x EE im (X, v X0), with

(dA,1)*x = 0 (1)

and

Ed(x)=jx whereEd=(lyd,l)*d"'. (2)

Here Ed is the `functional suspension' considered in (II. §11) of Baues [AH]. Using proposition (11.12.3)in Baues [AH] we get 43= iB a,a EE Ext(A,

(P*y=y*(4P+iBa) = y* ip + (Ex) * G,, a, 'B FO) where Ex is the partial suspension of x and where FO: M(A0, n) - M(B0, n) is the restriction of gyp. Since j(i B) 0 is trivial, we see that j (p * y = j cp * y = cp* jy and hence cp*(jy) does not depend on the choice of the realization ia. 11

If A is free abelian we know thatk in (6.15.11) is an isomorphism and hence F T,, (A, n) = 0 in this case. We are now ready to state the main theorem in this section.

(6.15.13) TheoremLet K be a CW-complex with dim(l:mK) < 3n - 1 and m, n >- 2 and let A be an abelian group. Then there is an exact sequence h A*,,

j7rm M(A, n) - 0.

Moreover we have for dim(ImK) < 3n - 2 the isomorphisms

F,K(A,n)=A®-rrm(Sn)

FTm (4,n)=-A*'7rm (S").

All morphisms are natural in A and K. The quadratic torsion products *' and *" are defined in Section 6.14. 236 6B MOORE FUNCTORS

We point out that the case dim(Y.mK) = 3n - 2 in the theorem is the first case outside the `quadratic range'. In this case n) is not quadratic in A and we have (6.15.14) n) = n)fordim lmK < 3n-2. Here L1 is the derived functor in the sense of Dold and Puppe [HN]. In fact the theorem has an extension to arbitrary dimensions of YmK by a spectral sequence of W. Dreckmann. The E2 -term of this spectral sequence consists of the derived functors Lj F, ( -, n), j >- 0, with n) = n). The spectral sequence converges to ir,, M(A, n) and is natural in M(A, n) and K. We now take K = S° in Theorem 6.15.13. Then we see by the inclusion (6.14.9), A *"lTm-,(S^) CA * A * 1Tm_t(S2"-1), that A *" lTm_{S"} = 0 is trivial for t n< 2n. Hence we obtain the following special cases.

(6.15.15) CorollaryFor n >- 2 one has the short exact sequences (k < 2n - 1) 0 -A(& lrkS" ->'TkM(A, n) -A * rrk_,(S") -0 0 -A(& ?T2n_,{S"} - lT2n- 1M(A, n) -A* ir2n_2(S") - 0 0 - r' (A) -> Tr2,,M(A, n) -A *'Tr2n_ 1(S") - 0 with r' (A) = A ® vr2n{S"} for n >: 3. Moreover one has the exact sequence -,r',"+'(A) A ir2"+1M(A,n) -*L1F,"(A) -*0. Here we have L1F,"(A) =A *'7r2 ,(S") and L2f,"(A) =A *"7r2n{S"} for n >- 3 andI', "+'(A) =A ®1T2n+,(S") for n>3. All sequences are natural in M(A, n) E Mn.

The naturality implies that the exact sequence of Theorem 6.15.13 induces a corresponding exact sequence for cross-effects. Restricting to the quadratic range dim(l'"K) < 3n - 2 we hence get the following corollary where I B, n) is the image of the map j: I B, n) -ir,n (M(A I B, n), M(A° I B°, n)) given by the pair EiA A 1B; see Definition 6.15.6 and (6.15.1)(5).

(6.15.16) CorollaryLet dim(!mK) < 3n - 2 with m, n >- 2 and let A and B be abelian groups. Then there is the following exact sequence of cross-effects of the functors in Theorem 6.15.13.

0-->Trp(A,B,_7rnxS2n 1S2n

d A ®B ® irnKS 2" -' --> irnKM(A I B, n) -* jirm M(A I B, n) -+ 0. 6B MOORE FUNCTORS 237

Here we use the formulas for quadratic cross-effects in (6.14.8); in particular Trp is the triple torsion product of Mac Lane; see (6.14.8)(5). We leave it to the reader to write down the cross-effect sequences for Corollary 6.15.15.

Proof of Theorem 6.15.13The proof relies on the exact EHP-sequence for mapping cones obtained in Theorem A.6.9. For this we use the fact that the Moore space M(A, n) = Cd is the mapping cone of a map d: X1=M(A1,n)-VXo=M(A0,n) where X1 = Y.X,' with X,' = M(A1, n - 1). The following commutative diagram extends the diagram in the proof of Theorem 6.15.10. The operator Em is (7rd,1)* d-' in Theorem 6.15.10.

kernel(d,1)*c 77m (X1 V X0)2

E, I K L ` TmK+1Cd 11 C 7rm+ I(CdI,XO)

Hm1 7rn 1(XXI' AX;)

I(X1vX0).Pm I11I 7r,

Here Pm_ I =[i I, i l - iod ]*is induced by the Whitehead product [i1,i1 -iod]: Y.X,'AX; -9X1 VX0 (1) where ie is the inclusion of X. in X1 V Xo for e = 0, 1. If dim(F,mX) < 3n - 2 then the kernel of Em is the image of Pm. For dim(Y,mX) = 3n - 2 the kernel of Em is the image of

Pm: Xj' AX,' VX1)2- 7r1K (XI VXo), (2) with Pm =([i1,iI -iod], -i1)*. Hence we get by (6.15.11)

I T,, (A, n) =Em kernel(d,1)* = kernel(d, 1),, /image Pm. (3) This formula can be used to compute the functor n) in Theorem 6.15.10. In fact for dim(ImK) < 3n - 2 we can identify (d,1)* with dl as in the proof of (6.15.8); similarly we can identify Pm with d2 in Definition 6.14.7. This shows

n) =A *'7rm (S")fordim(YmK) < 3n - 2. (4) 238 6B MOORE FUNCTORS

Finally an easy diagram chase yields the exact sequence in Theorem 6.15.13 since the row and the column of the diagram above are exact. In fact the map e is induced by Em and h is induced by Hm since the kernel of Pm-I =d2 is

kernel Pm_ 1 =A = image Hm (5) by Definition 6.14.7(7). Moreover 6 is induced by

d: image Hm -> cokernel(d,1) in the diagram where the cokernel of (d,1)* is F,n (A, n). The maps A and j are considered in (6.15.11).

Theorem 6.15.13 is slightly more general than the corresponding result (9.5) in Baues [QF] where we deal only with the quadratic part of Theorem 6.15.13. For the classification of 1-connected 5-dimensional homotopy types in Chapter 12 we also need the non-quadratic part of Theorem 6.15.13. THE HOMOTOPY CATEGORY OF (n-1) -CONNECTED (n + 1) -TYPES

We have to consider the hierarchy of categories and functors (n >- 2)

types.' types.' FP-- types F-- (1)

where types is the full category of (n - 1)-connected (n + k)-types, that is, of CW-spaces Y with 7riY = 0 for i < n and i > n + k. The functor P is the Postnikov functor which carries an (n + k)-type to its (n + k- 1)-type, k >- 1. Since (n - 1)-connected n-types are the same as Eilenberg-Mac Lane spaces K(A, n), we can identify them with abelian groups. In fact one has an equivalence of categories, see (6.1.1),

ir,,: types.' -a Ab. (2)

From this point of view (n - 1)-connected (n + k)-types are natural objects of higher complexity extending abstract abelian groups. Following up this idea J.H.C. Whitehead looked for a purely algebraic equivalent of an (n - 1)- connected (n + k)-type, k > 0. An important requirement for such an algebraic system is `realizability' in three senses. In the first instance this means that there is an (n - 1)-connected (n + k)-type which is in the appropriate relation to a given one of these algebraic systems, just as there is an Eilenberg-Mac Lane space K(A, n) whose nth homotopy group is isomorphic to the given abelian group A. The second kind is the realizability of morphisms between such algebraic systems by maps between the corresponding (n + k)-types, and the third kind is the uniqueness up to homotopy of such realizations of a given morphism. Thus we are searching for a category C of algebraic models equivalent to the category types' as achieved in (2) above for k = 0. Given such a category C the computation of bype and kype functors on C would give us then algebraic models of homotopy types by use of the classification theorem in Chapter 3. J.H.C. Whitehead classified the objects in types' by homomorphisms

r,,' (A) --, B. (3)

A suitable category of algebraic models, equivalent to types;,, however, is not given in the literature. Using results on the category Mn of Moore spaces of degree n we shall describe such algebraic categories. This will be important for the classification of (n - 1)-connected (n + 3)-dimensional homotopy types, n 2- 2, in the following chapters. 240 7 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA 7.1 A linear extension for typesn'

We show that the homotopy category types' (n - 1)-connected (n + 1- types can be described as a linear extension of the category I'Ab" consisting of quadratic functions for n = 2 and of stable quadratic functions for n >- 3. Recall that a function q7: A -b B between abelian groups is quadratic if 71(-a) = rl(a) for a c =A and if [a, b],, = q(a + b) - -q(a) - -q(b) is bilinear in a, b EA. The universal quadratic function y: A - I'A has the property that there is a unique homomorphism° : FA -> B with q ° y = q. This way we identify a quadratic function 71: A -. B and a homomorphism q ° : TA - B. Let I'Ab be the category of quadratic functions. Objects are quadratic functions and morphisms cp = ((po, cpl): r, -> q' are pairs of homomorphisms for which the diagram A Vo -A'

17 117, B - B' commutes or for which equivalently (71') ° F(cp0) = cplq 0. A quadratic function 71: A -> B is stable if [a, b],7 = 0 for all a, b E A, that is, i1 is a homomorphism with q(2 a) = 0 so that a stable function 71 can be identified with a homomorphism q': A ®1 /2 - B. Let SI'Ab = TAb" (n >- 3) be the full subcategory of I'Ab = FAbz consisting of stable functions. We have the full inclusion Ab c FAb which carries the abelian group A to the universal quadratic map yA: A - I'A. We also have the full inclusion Ab c SI'Ab which carries A to the universal stable quadratic map o-yA: A -A ®1 /2. Here oyA is the quotient map. (7.1.2) Remark The functor F,,: Ab - Ab is given by I'z = IF and F,,' = ®1 /2 for n z 3. The Grothendieck construction of the bifunctor Ab°P X Ab - Ab is the following category. Objects are homomorphisms f: B with A, B E Ab and morphisms (cpo, (p, ): f -p g are pairs of homomorphisms in Ab with gT `(cpo) = cp, f. There is a canonical isomorphism of categories FAb" = Gro(Hom(F,,,-)) which carries the quadratic function 17 to Let 'r E ir3S2 be the Hopf map and let ii" E iri+,S" be the (n - 2)-fold suspension of 71. Then for any space X in Top* the induced function 71,*,: it"(X) - ir,,+,M, 17, (a) = a - n", is quadratic; moreover q,*,is stable for n > 3. Since rl,*,is natural in X we thus obtain the functor k" : types;, --> FAb" 7 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA 241

which carries an (n - 1)-connected (n + 1)-type X to 71,* and which carries a map f : X --> Y in types;, to the induced map (a (f ),'n + ,(f )) between homotopy groups.

(7.13) PropositionThe functor k is a detecting functor, n >- 2.

Proof For X in types' let A = ir X and B = 7r+ I X. Then the Postnikov decomposition of X shows that X is the fibre of a classifying map kx: K(A,n)-->K(B,n+2) which is the first k-invariant of X. The homotopy class of kX is an element kXE [K(A,n),K(B,n + 2)] =Hn+2(K(A,n),B)=Hom(F, (A),B) and thus kx corresponds to a quadratic map A -> B which actually is ij,*. The second isomorphism is obtained by the universal coefficient theorem for cohomologysince we haveisomorphisms (n >- 2)H,, K(A, n) = A, H,,+ IK(A, n) = 0, and Hi+2 K(A, n) - F,, (A). Hence each quadratic map 7) E Hom B) is realizable by a space X in types;, with classifying map kx = 71. Moreover each morphismgyp: kx - ky in I'Ab corresponds to a homotopy commutative diagram K(A,n) K(A',n) Ikr kX1 w, . K(B, n + 2) K(B', n + 2) which thus yields a principal map between fibre spaces,gyp: X --> Y, which realizes gyp. 11

(7.1.4) Remark In Definition 2.5.8 we describe a detecting functor (n >t 2)

A: types' -* which by the identification of categories in Remark 7.1.2 coincides with the detecting functor k in Proposition 7.1.3 above; see also Theorem 6.4.1.

(7.1.5) DefinitionFor each abelian group A we have the Eilenberg-Mac Lane space K(A, n). Extending this notation we introduce quadratic spaces n) as follows. Let n >: 2 and let q: A - B be a quadratic function which is stable for n3. Then we write X = K61, n) if X is an (n - 1)-connected (n + 1)-type for which isomorphisms A =- ir,, X, B = 1 X arefixed such that n"-Ir"+IX=B q:A=7r,, X 242 7 (n - 1)-CONNECTED (n+ 3)-DIMENSIONAL POLYHEDRA coincides with rl,*. The proposition above shows that the homotopy type of K(rl, n) is well defined by 71. Moreover each morphism cp = (rpo, rp,):n -> 71' in I'Ab has a realization ;p: K(,q, n) --> K(-q', n) with rp. Here is not uniquely determined by gyp. The loop space of an Eilenberg-Mac Lane space is cIK(A, n) = K(A, n - 1), n >- 2. Similarly we have for n z 3 SZK(,1, n) = K(,q, n - 1) where q is stable. Hence K(q, 2) is a loop space if and only if i is stable. In the next result we classify maps.

(7.1.6) PropositionLet rl: A -> B and r)': A' - B' be quadratic functions which are stable for n >: 3. Then we have for n >: 2 the exact sequence

Ext(A,B') N k This is an exact sequence of abelian groups if q' is stable. For n = 2 the group Ext(A, B') acts freely on the set [K(i, 2), K(-q', 2)] with orbits given by k2. Moreover for n > 2 we have the linear distributivity law

(+1r+ /3)(cp+ a) = a/np+ q*(a) + (p*(j3). Here is the composite

K(q,n)-- K(7',n)-' K(71", n) and a e Ext(A, B'), 0 E Ext(A', B") and we set 4* = (/r,)*, rp* _ (cpo)*.

We point out that the loop space functor Cl is compatible with the exact sequence in the proposition. Therefore we get a sequence of functors: (7.1.7) types' 2 types' types' where Cl on types' is full and faithful and where Cl: typesn - types,',-, is an equivalence of categories for n >_ 4. Hence we see that types;, for n 3 is equivalent to the full subcategory of typesZ consisting of all K(17, n) for which iis stable. Moreover the proposition shows that we have a linear extension of categories

(7.1.8) E >- types;, fl.. rAb,, where E is the bimodule on FAb given by E(q,,U') = Ext(A, B') for q': A -' B, n: A' - B'. The functor k,, carries K(77, n) to 77. For the group CE(K(r1, n)) of homotopy equivalences of the space K(q, n) we obtain by (7.1.8) the extension of groups (7.1.9) Ext(A,B)-C(K(i1,n))-»Aut(n) 7 (n- 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA 243

where Aut(rl) is the group of automorphisms of the object 17 in FAb with Aut(,q) c Aut(A) x Aut(B). The extension (7.1.9) is split if A is cyclic or Ext(A, 77/2 = 0.

Proof of Proposition 7.1.6 We have the fibre sequence (n >_ 2) K(B',n+1)-K(q',n) PK(A',n)K(B',n+2) and the action µ: K(,q',n)xK(B',n+1)- K(i',n) on the fibre. For X = K(,q, n) we thus obtain the action A* : [X, K(rl', n)l X H" +'(X, B') ->[X, K(ij', n)] which carries (x, a) to µ*(x, a). The projection p: X = K(11, n) --* (A, n) induces the inclusion (see for example (3.3) in Baues [MHH]) p*: Ext(A, B') = H"+'(K(A,n), B') N H+'(X, B') so that we can define the action in the proposition by x+a=µ*(x,p*a)for aE Ext(A,B'). By (V.10.7)-(V.10.9) in Baues [AH] we see that all elements in [K(rl, n), K(rl ', n)l = PRIN(ij, r1') are given by principal maps between fibre spaces. Therefore the proposition is a very special case of (V.10.19) in Baues [AH].

For an abelian group A we have the universal quadratic function yA = YA: A -> FA and the universal stable quadratic function QyA = -y;: A -A ®1 /2 which is the quotient map, n >_ 3. The (n + 1)-type of a Moore space M(A, n) is K(y,;, n) for n z 2 and we have p"+i: M(A,n) -+P"+IM(A,n)=K(y" ,n) inducing the isomorphisms 7r,, M(A, n) =A and zr"+,M(A, n) = r(A). The Postnikov functor P"+, yields a full embedding of homotopy categories

P,,, : Mn ctypes;, which is compatible with the linear extension in (7.1.8) and Section 1.3. That is, for n 2- 2 we have the commutative diagram of linear extensions

Ext(-, N Mn Ab (7.1.10) n n n E types;, where the full inclusion Ab c I'Ab" carries A to y,; . In fact, we can describe 244 7 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA the category types' completely in terms of the category M"; for this we introduce the enriched category of Moore spaces in the next section.

(7.1.11) Remark For a CW-space X and n >- 2 we have the natural isomorphism of groups H"(X, A) = [X, K(A, n)] where the left-hand side is the reduced singular cohomology with coefficients in A and where [ X, Y ] denotes the set of homotopy classes of maps X --> Y. Now let i be a stable quadratic function, that is a homomorphism 17: A ®1 /2 -' B. Then we define for a CW-space X the cohomology

H"(X, r7) = [X, K(1, n)1,n >- 2, with coefficients in q. This is an abelian group and a functor in X. The fibre sequence for K(i, n) yields the following exact sequence of abelian groups where SqA: Hm(X, A) -' H'°+2(X, A ®1/2) is the Steenrod square (which is trivial for m = 1): n.sg ,H"+1(X,B) H"-1(X,A) -H"(X,77) >H"(X,A)*7.s A,H"+2(X,B). One can check that q* SqAis also induced by the map 77: K(A, m) --> K(B, m + 2) given by 17. The cohomology H"(X, q) is not functorial in r7 but for 9: q - r)' a map gyp: K(77, n) --> K(17', n) induces a homomorphism ilp * : H"(X,,y7) --> H"(X, i7') which is not well defined by gyp. We clearly have H"(K(ij, n), q') = [K(r), n), K(i1', n)]. This group can also be computed by the proposition above. An algebraic description of this group is given in Theorem 7.2.9 below. The cohomology groups H"(X, 71) were recently studied in quite different terms by Bullejos, Carrasco, and Cegarra [CC].

7.2 The enriched category of Moore spaces The full homotopy category M" of Moore spaces M(A, n) is studied in Chapter 1. We here use the category M' to obtain an algebraic description of the category types;, of (n - 1)-connected (n + 1)-types, n >- 2. The linear extension of categories

(7.2.1) Ext(-,F,)NM"- Z Ab is given by the homology functor H" and by the action + of Ext(A, F, (B)) on the set of homotopy classes [M(A, n), M(B, n)] where 7r"+IM(B, n); 7 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA 245 compare Section 1.3. We use the action in the following definition of the `enriched' category of Moore spaces.

(7.2.2) DefinitionThe enriched category TM" of Moore spaces is defined as follows (n >- 2). Objects are pairs (M(A, n), 77) where77: A -+ B is a quadratic function which is stable for n > 3. A morphism

{(PI _Fo}: (M(A,n),i1) -> (M(A',n),7)') in FM', with 77': A' -* B', is represented by a tuple API E Hom(B, B') E Ext(A, B') ;Po E [M(A, n), M(A', n)] where ;Po induces po: A - A' in homology such that (c, (p,):17 - 71' is a morphism in I'Ab, that is n' cpo = cp, r7. The equivalence class {(ps, ", ;30) is given by the equivalence relation forSE Ext(A,F'A'). Here we use the homomorphism r7': ,,A' - B' given by r7' and we use the action + of Ext(A, F, A') on the set [M(A, n), M(A', n)] which yields ;Po- 8. We define the composition in FM" by

1011 I1 f 4,(pi,01+P0* o'Po} Here trio ipo is the composition of maps between Moore spaces. For n >: 3 we obtain an additive structure of FM" by the formula

This shows that FM" is an additive category for n >: 3. Moreover the suspension yields the canonical isomorphism FM" _- I M"+' for n >- 3. We have the full inclusion of categories (7.2.3) M" c FM" which carries M(A, n) to (M(A, n), y,,) where y,,: A - F'A is the universal quadratic map (stable for n >- 3). Moreover the inclusion carries a map 'Po in M" to the map (f((po),0, fPo} in FM". The linear extension (7.2.1) has the following generalization for the category I'M".

(7.2.4) Lemma There isa linear extension of categories (n >- 3) E-+FM" k FAb" where E is the bimoduleon I'Ab given by E67,10 = Ext(A, B') as in(7.1.8). Moreover k" is the forgetful functor which carries the object (M(A, n), 17) to 17. 246 7 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA

T h e action + o f E i s given i n the obvious w a y b y ((pl, , 4 p o) + a = (cpI, + a, The lemma is readily obtained by use of the linear extension (7.2.1). Moreover a cocycle for the extension (7.2.1) yields a cocycle which determines the extension I'M" in the lemma. (7.2.5) DefinitionLet spaces" be the full subcategory of (n - 1)-connected CW-spaces in Top*/~. We define a functor (n >- 2) K: spaces" 1' M" as follows. For an (n - 1)-connected CW-space X let A = H" X = 7T,, X and let 71= 71,*,: A = 7T" X -> ir"+,X = B be induced by the suspended Hopf map. Then K" carries X to the object (M(A, n), ii). We now choose for each X a map a: M(A,n)->X (1) which induces the identity H"(a) of A = H" X. For a map F: X --' X' between (n - 1)-connected CW-spaces we thus obtain the diagram M(A, n) -

WoIl I 111F (2) n) X'IF M(A', where ;Po realizes the homomorphism cpo = H" F: A --.,-A'. Thus the diagram commutes if we apply the homology functor H, but in general cannot be chosen such that the diagram actually commutes in Top*/=. Let ii': A' = H,, X' _ X' -->B = it,, ,X' be given by X'. Then Ext(A,B') (3) is determined by the universal coefficient sequence for [M(A, n), X'] _ 77"(A, X'). We now define the functor K,, on F: X - X' by K,,(F) = ( p,,0(;?o), ;o): (M(A',n),77') (4) where cp, = -rr"+,(F). (7.2.6) LemmaK,, is a well-defined functor. Proof The definition of K,,(F) depends on the choice of iPo. A different choice is of the form iGo - S with S E Ext(A, F, A'). Now we get O(;po-8)=0-'(Fa-a(fP"- S)) =O(ilpo)+'q*S. This shows that we have an equivalence

(Bp1,0(450),;po)-('PI,0(;Po-6),oo-8) 7 (n-1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA 247

and therefore K,,(F) is well defined. Now it is easy to check that K" is a well-defined functor which clearly depends on the choice of the maps a above.

We now restrict the functor K" to the category of (n - 1)-connected (n + 1)-types.

(7.2.7) TheoremThe functor K" above yields for n >_ 2 an equivalence of categories

K,,: types;, -a FM".

Proof The functor Kn induces a map between linear extensions of categories

k, EH types;, I'Abn

II E N I'M" - I Ab,,

This implies that Kn is an equivalence of categories.

The theorem shows that we can use all results on the category M" for computation in the category types;,. In particular since we have algebraic models of M" we obtain in this way algebraic categories equivalent to the category types,',. For example we have for n > 3 the equivalence of categories M" = G where G is the algebraic category in Section 1.6. Using this equivalence we obtain in thesame way as in Definition 7.2.2 theenriched category FG as follows.

(7.2.8) DefinitionRecall that we have for each abelian group A the extension in Ab

A ®Z/2HG(A) -*A*7L/2

associated with A *71/2 cA -A ® 71/2. Objects in the enriched category 1'G are stable quadratic functions 17: A -B or equivalently homomorphisms rt: A ®7L/2 -+ B with A, BE Ab. Let r,': A' -> B' be afurther object in FG. A morphism

11, (1) C, 'Po' ;Po}: rl --' 248 7 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA in rG is represented as follows. Let C E Ext(A, B') and letgyp,, go, 4P0 be homomorphisms in Ab for which the following diagram commutes.

B 77 A(& Z/2 - G(A) -. A* Z/2 I P) Iwo®I IiGo IZ/2 (2) B' F- A' ®7L/2 » G(A') A'* n' µ

Then the equivalence class ( g1, , go, iPo) isgiven by the relation

(3) for S E Ext(A, A' (&7L/2) = Hom(A * Z/2, A' ®7L/2). Composition is defined by

{gi, , go' ;00) o {Qi, go, iPo} = (91pl,(91) *+ go go, 4P0;000 Also we obtain the structure of an additive category for rG as in Definition 7.2.2 by

Recall that SFAb is the category of stable quadratic functions. We obtain as in (7.1.8) a linear extension of categories.

+ E rG:SrAb. (4) Here 0 is the identity on objects and carries (1) to (g1, c'0): rt - if in SrAb. The action of E is given for a E Ext(A, B') = E(-q, 70 by

( a, go,iPo). One readily checks that (4) is a well-defined linear extension. We derive from Theorem 7.2.7 the next result which provides us with an algebraic model of the category types' for n -- 3.

(7.2.9) TheoremFor n > 3 one has equivalences of additive categories types'=rM' =rG which are compatible with the linear extensions in the proof of Theorem 7.2.7 and Definition 7.2.8(4).

Using the equivalence we identify a map K(q, n) --> K(rl', n), n >: 3, with an algebraic morphism r) -> if inG. 8 ON THE HOMOTOPY CLASSIFICATION OF (n - 1) -CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA, n>-4

J.H.C. Whitehead in 1948 classified (n - 1)-connected (n + 2)-dimensional homotopy types. Since then it has been a challenging problem to consider the next step in the classification of (n - 1)-connected (n + 3)-dimensional homotopy types. Various authors have worked on this problem in the stable range n > 4, for example Shiraiwa [HT], Uehara [HT], Chang [PI], [HT], [AS], [NH], and Chow [HG]. They use a complicated method of primary and secondary cohomology operations; compare Section 8.5 below. We here apply the new method of boundary invariants which, via the classification theorem 3.4.4, yields fairly simple classifying data for (n - 1)-connected (n + 3)-dimensional homotopy types. In this chapter we deal only with the stable case n > 4. In Chapters 9 and 12 we describe the considerably more intricate unstable cases, n=3 and n=2.

8.1 Algebraic models of (n- 1)-connected(n + 3)-dimensional homotopy types,n >- 4

We introduce the purely algebraic category of A3-systems. Then we formulate the main result of this chapter which shows that A3-systems are algebraic models of certain homotopy types. Recall that we have for an abelian group A the short exact sequence

(8.1.1) A®Z/2NG(A)-.A*71/2

associated with

TA:A*Z/2HA-.A®71/2.

Here i is the inclusion andp is the projection. Theabelian extension (8.1.1) is determinedup to equivalence by the property A-1(2 A- '(x)) = TA(x) for x (H A * 71/2. For each homomorphism gyp: A --* B there is a homomorphism : G(A)- G(B) compatible with A and µ in(8.1.1), that is j. = (gyp *71/2)A 250 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA and A ((p (9 1/2) = ;pA. We have the dual extension, see Lemma 8.2.7,

(8.1.2)

Ext(A,71/2)>' >Hom(G(A),Z/4) Hom(A,Z/2)

II II II Hom(A*Z/2,71/4) - Hom(G(A),7L/4)-> Hom(A®1/2,1/4)

Here the bottom row is obtained by applying the functor Hom(-,1/4) to extension (8.1.1). The isomorphism at the left-hand side is given by Lemma 1.6.3. We now use the extension (8.1.1) and (8.1.2) for the definition of the following extensions G(q) and G(A,,q) respectively.

(8.13) (A) DefinitionFor a homomorphism q: H ®1/2L in Ab let G(71) be defined by the push-out diagram in Ab

L ®1/2 >A+ G('q) - H * 1/2

17011 push Iii II H®1/2 » G(H) -» H*1/2 where the bottom row is given by (8.1.1). Hence G(rl) in the short exact top row is the abelian extension associated with the homomorphism

TcfTH:H*1/2cH-.H®1/2-- L*1/2cL -.L ®1/2.

(8.13) (B) DefinitionRecall that objects in the category G are abelian groups A and morphisms ;P = (gyp, ;p): A -> B are pairs

((p, irp) E Hom(A, B) X Hom(G(A),G(B)) compatible with A and µ in (8.1.1); see Section 1.6. Next let

srAb' c sfAb be the full subcategory of stable quadratic functions 71: H ®1/2 - L for which there exists a factorization77: H 0 1/2 - G(H) - L; see (7.1.1). 8 (n- 1)-CONNECTED (n+ 3)-DIMENSIONAL POLYHEDRA 251

Morphisms (ip a/ro):71 -* r)' with rt': H' ® 71/2 -* L' are pairs (iJr,, R(io) E Hom(L, L') ® Hom(H, H') compatible with ri and if. We define an algebraic functor G:G°PxSI'Ab'->Ab as follows. If A or H is finitely generated let G(A, rl) be given by the push-out diagram in Ab

Ext(A,L) ° ' G(A,77)µ °° Hom(A,H(9 71/2)--*0

a.1 push

Ext(A,H(& 71/2) Ti.

II Ext(A, 71/2) ® H -* Hom(G(A), 71/4) ® H -- Hom(A, 71/2) ® H 0

Here the bottom row is obtained by applying the functor -®H to the extension (8.1.2). Induced homomorphisms are defined by

(cp, ;p)*= Ext((p,L)®Hom(;p,71/4) ® H (.#i,4o)* =Ext(A,01)ED Hom(G(A),Z/4)®1Iro.

(8.13) (C) AddendumIn the general case we have to use the following more intricate definition of the group 6(A, 71) which canonically coincides with the definition (8.1.3) (B) if A or H are finitely generated. For n z 0 let 7L/n[H] be the free 71/n-module generated by the set H. We have the canonical map p,,:71/n[H]H®71/n which carries E; a;[x;] witha; E 71/n, x; E H to thecorresponding sum E; x; ® a,. Clearly p is a surjective homomorphism which is natural in H E Ab. Let K(H) be the kernel

P2 K(H) = kernel{71/4[ H ] P-* 71/2[ H ] H ® 71/2} where p is reduction modulo 2. Naturality shows that K is a functor Ab - Ab. We now define the natural transformation

OH : K(H) * 7L/2 -> cok(rf ) 252 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA where TH is defined as in (8.1.1). For y = E, a;[x;] E K(H)*7L/2 with a, (-=Z there is x E H such that E, a;x, = 2x in H. Now 0 carries y to the element in cok(TH) represented by x. One readily checks that 0 is a well-defined homomorphism natural in H E Ab. In fact, OH is obtained by the following commutative diagram

K(H) H 71/4[H] PzP 10, 10, cokTHNH®7L/4 -. H®7L/2 in which the bottom row is short exact. The map 0' carries [h] to h ® 1 and the restriction of 0" to K(H)*7L/2 is OH. For abelian groups A, H we define A * Ext(A, H 0 71/2) by the image of

A*:Ext(A,H(& 71/2)-->Ext(A,G(H)) induced by A in (8.1.1). Clearly A * Ext(A, H (& 71/2) is functorial in A, H E Ab. If K is a 1L/2-vector space we get

A * Ext(K, H ®7L/2) = Hom(K, cok TH ).

Using K = K(H) * 1/2 and OH above we thus obtain a homomorphism

0: Hom(A*Z/2,K(H)*7L/2)- A*Ext(A,H®71/2) 0(a)=a*OH which is natural in A and H. We are now ready to define 6(A, 71) by the push-out diagram:

Hom(A*7L/2,K(H)*7L/2) N Hom(G(A),7L/4[H]) -. Hom(A®71/2,7L/4[H])

8! II 0*Ext(A,H®71/2) push Hom(IA,7L/2[H])

Ext(A,L)% G(A,71)µ ., Here 71* is well defined since 77 factors through A: H ® Z/2 -> G(H). The inclusion j carries a to the composition j(a): G(A) -A*71/2--K(H)*71/2c7L/4[H]. 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA 253

We observe that image(j) = kernel(P2) * 0*, so that the bottom row is short exact since (p2)* A* is surjective. Induced maps on G(A, r1) are now defined by (cp, ip)* = Ext(cp, L) ® Hom(4p,71/4[H1) (+6i,#0)* = Ext(A,+G,) ® Hom(G(A),71/4[4rol). This completes the definition of the functor G in Definition 8.1.3 (B).

Using the notation on the groups G(71) and G(A, 71) in Definitions 8.1.3 we are now ready to define algebraic models of (n - 1)-connected (n + 3)- dimensional homotopy types which we call A3-systems.

(8.1.4) Definition An A3-system

S=(Ho,H,,H3,ir1,b,,r1,b3,/3) (1) is a tuple consisting of abelian groups H0, H H3, rr, and elements b2 E Hom(H2, Ho (9 Z/2),

11 E Hom(H0 (9 71/2, rr, ) (2) b, E Hom(H3,G(r1)), /3EG(H7,i1#).

Here 71., = gO(71® 1) is the composition

G(11) r,#: Ho ® 71/2710I1rrt ®Z/2 -* cok(b3) (3) where q is the quotient map for the cokernel of b3. These elements satisfy the following conditions (4) and (5). The sequence

H2 ----+ Ho (9 Z/2 77 (4) is exact and 03 satisfies µ(0) = b2 (5) where µ is the operator on G in Definition 8.1.3. A morphism

(cpo, 92, 93, (Pn, (P F): S -S' (6) between A3-systems isa tuple of homomorphisms

q: H; --, H; (i = 0, 2,3) cp,,: rr, -> >r, 9r: G61) -G(r1') 254 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA such that the following diagrams (7), (8), (9) commute and such that the equation 10 holds. H2 b * Ho ® 71/2 -- zr,

1Vz 1(P0®1 1lpr (7) H' Ho ® 71/2 -1 ir, V

ir, ® 71/2NG(7) --o Ho *71/2

1 (8) 1rj ® 7/2 » G(-1') -µ" Ho' * Z/2 H3 -, G(q)

1Vr (9) H3bG(7) Hence 'pr induces 'pr: cok(b3) -> cok(b3) such that g067 (9 1) -p g0(77' 0 1) is a morphism in SFAb' which induces (pr)* as in Definition 8.1.3. We have OPo,Qr)*(/3(p2.'p2)*((10) in G(H2, g0(7j' ® 1)). In (10) we choose 'P2 for '02 as in (8.1.1). The right-hand side of (10) does not depend on the choice of ;D2.

An A'-system S as above is free if H3 is free abelian, and S is injective if b3: H3 -+ G(q) isinjective. Let A3-System resp. A3-system be the full category of free, resp. injective, A3-systems. We have the canonical forgetful functor 0: A3-System -A 3 -system (11) which replaces b3: H3 - G(q) by the inclusion b3(H3) c G(,7) of the image of b3. One readily checks that this forgetful sk is full and representative.

(8.1.5) Definition We associate with an A3-system S as in Definition 8.1.4 the exact I'-sequence

H3 b, G(-l) 'r2 H2 Ho (9 7L/2 7 1r1 - H, - 0. Here H, = cok67) is the cokernel of r7 and the extension cok(b3) > a2 - ker(b2) (1) 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONALPOLYHEDRA 255

is obtained by the element 63 in Definition 8.1.4,that is, the group 7T2 = Tr( (3t) is given by the extension elementRt E Ext(ker(b2),cok(b3)) defined by

Rt=A-1(j, j)*(,8) (2)

Here j: ker(b2) c H2 is the inclusion. Theelement Rt does not depend on the choice of (j, j) in G. Compare (2.6.7). Recall that spaces,', denotes the full homotopycategory of (n - 1)- connected (n + 3)-dimensional CW-spaces X andthattypes,2,isthe full homotopy category of (n- 1)-connected (n + 2)-types. We have the Postnikov functor

P: spacesrt -+ typesn

which carries X to its (n + 2)-type.

(8.1.6) TheoremIn the stable range n >_ 4 there are detecting functors:

A': spaces -A3-System A': types -A3-system.

Moreover there isa natural isomorphism 4A(X) = A'P(X) for the forgetful functor 0 in Definition 8.1.4(11) andfor the Postnikov functor P above.

Remark Theorem 8.1.6 isan application of the detecting functors A', A' in the classification theorem 3.4.4.These detecting functors are defined by boundary invariants. Thereshould also be a similar result to Theorem 8.1.6 above concerning the detecting functors A,A in Theorem 3.4.4 given by k-invariants. Itturns out, however, that the computation of boundary invariants is simpler than the corresponding computation of k-invariants. The analogue of Theorem 8.1.6 for k-invariants remainsan open problem though for ir,,+ I = 0 we have discussed the functors A, A in Theorem 3.6.5; for 'r,, + i = 0 Theorem 8.1.6 above corresponds exactly to the detecting functor A' in Theorem 3.6.5. The result therewas based on the computation of n) and K(B, n)). In Theorem 8.1.6 we treat the case 'r + 1 X * 0 whichisbased on the computation of H + 3 K(q, n) and r+ 1(A, K(i, n)) in Section 8.3 below.

Let S bea free A3-system. The detecting functor A' in Theorem 8.1.6 shows that thereis a unique (n - 1)-connected (n + 3)-dimensional homotopy 256 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA type X =Xs, n >: 4, with A'(X) = S. Then the F-sequence for S is the top row of the following commutative diagram

(8.1.7)

H3 - G('q) - H2 7T2 -'Ho ®Z/2-'-7-* 'iT 1 -'H,

II II at II II II II kin +3X - r"+2X -i"+2X-H"+2X- I'"+1X --' 1r"+1X -" H"+tX

The bottom row is Whitehead's certain exact sequence for X. The diagram describes a weak natural isomorphism of exact sequences. This shows that the homotopy group it"+2(X) is completely understood in terms of the A3- system S.

(8.1.8) Example For an abelian group A let SA be the unique A3-system with Ho =A and H, = H2 = H3 = 0. That is

_Ho, H2, H3, ir1, b2, r, b3,,8 SA A,0,0,A®7L/2,0,1,0,0

Then the space X with A'(X) = SA is the Moore space X = M(A, n), n > 4.

As an application we now derive from Theorem 8.1.6 the following result on maps into spheres.

(8.1.9) TheoremLet Xs be the (n - 1)-connected (n + 3)-dimensional homotopy type associated with the A3-system S in Definition 8.1.4 with n >: 4. Then a homomorphism po: Ho = H"(Xs) -> H"(S") = 7L is realizable by a map Xs -+ S" if and only if there exist homomorphisms

gyp,,: i1 = rr"+1(Xs) -3 l/2 and0r: r(ui) = r"+2(Xs) -* Z/2 such thatq''i = cpo ® 71/2, (cpo (9 1)b2 = 0,cprO = cp ® 1,(Prb3 = 0, and ((Po, rpr)*( G) = 0.

Proof The conditions show that

('po 1010, gyp,,, , (Pr): S -> Sz is a morphism between A3-systems which is realizable by a map Xs -> S" since A' in Theorem 8.1.6 is a detecting functor; see Example 8.1.8. Remark The problem treated in Theorem 8.1.9 has an old tradition in algebraic topology, starting with a theorem of Hopf. Many authors considered 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA 257

maps from an (n + k)-dimensional polyhedron into the sphere S" for k = 0, 1, 2. An explicit criterion like in Theorem 8.1.9 for k = 3 was not achieved in the literature. Clearly Theorem 8.1.9 is only a simple application of the classification theorem 8.1.6 since more generally this theorem can be used to decide what homology homomorphisms H,, (Xs) - H,, (Xs,) are realizable by a map Xs -*Xs..

The stable nth homotopy group of a space X is the direct limit llm{Ir,, X - 7rn+11 X -' ITn+2y2X ... } given by the suspension homomorphism 1: 7Tn+k("X) - Trn+k+1(y.k+'X) for k > 0. As a simple application of Theorem 8.1.6 and (8.1.7) we now obtain a well-known result; see for example G.W. Whitehead [RA]:

(8.1.10) PropositionThe third stable homotopy group of the real projective space R P. is ir3 (If8 P) =71/8.

This result follows also from the proof of the well known Kahn-Priddy theorem. For the proof we consider the following explicit example of an A3-system; see also Section 12.6 below.

(8.1.11) Example Ler R P4 be the real projective 4-space. Then the space X=-'RP4 withn>-4, (1) is an (n- 1)-connected (n + 3)-dimensional complex forwhich the A3-system A'(X) = S is given by

H0, H2,H3,ir 1,b2,r7,b3,f3 S (2) (71/2,71/2,0,71/2,0,1,0,0(1) Hence S determines the homotopy type of Xs = X = 1,n-1R P4. We derive S from the following facts. It is well known that the homology groups of X are k X = 71/2 = H"12X and that H, X = 0 otherwise, i > 0. Since the generator Y E H2(R P4, 71/2) has a non-trivial cup square y U y E H4(f8 P4, 71/2) the space X has a non-trivial Steenrod square. Hence X is not a one-point union of Moore spaces. Moreover, for the real projective 3-space R P3 we know, for n>3,

In-1RP3=M(7Z/2,n)VSn+2 (3) compare (IV.A.11) in Baues [CH]. This does not hold for n = 2. We now prove (2). By (3) we know that b2 = 0. Since H1 = 0 we obtain ?r, = Z/2 and r1= 1. Hence G(i7) = 71/4. Since H3 = 0 also b3 = 0. Now f3 is in theimage of A: Ext(H2, F(l1)) = Z/2 - r(H2, A(i7 (& 1)) 258 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA since µ( 0) = b2 = 0. Moreover /3 9L 0 since X is not a wedge of Moore spaces. Hence /3 = .(1). Therefore the extension in Definition 8.1.5(1) G(i)=71/4H7re-.H2=Z/2 is non-trivial and thus yr"+ 2X = Z/8. This proves Proposition 8.1.10.

8.2 On irn+2M(A, n)

Moore functors are dual to the Eilenberg-Mac Lane functors; see Chapter 6. While many computations on Eilenberg-Mac Lane functors can be found in the literature there is only a little known on Moore functors. In this section we describe explicit examples of Moore functors which are needed for the proof of the classification theorem 8.1.6. We compute the homotopy groups 'fin 12 M(A, n) and am + 1(A, M(B, n)) of Moore spaces and we determine the functorial properties of these groups, n >- 4. Let M' be the full homotopy category of Moore spaces M(A, n). Mor- phisms in M' are homotopy classes of maps -ip: M(A, n) - M(B, n). For n >- 3 we have the algebraic category G which is equivalent toMn. Objects in G are abelian groups and morphisms A -> B are proper homomorphisms ((p, ii): G(A) - G(B) where G(A) is part of the extension (8.1.1); compare Section 1.6. There is for each A an isomorphism, compatible with 0 and (8.2.1) G(A) = 7r"(71/2, M(A, n)); see (1.6.4). Using this isomorphism we obtain the equivalence of categories

(8.2.2) M- :;G G forn>: 3 which carries M(A, n) to A and carries ; to (gyp, i/r) with qp = H" P and ,,= ar"(71/2, gyp). Hence a proper homomorphism (gyp, ur): G(A) - G(B) determines a unique element ipE [M(A,n),M(B,n)] with H"fp = cp and lr"(71/2, ip) We use (8.2.2) as an identification of categories. The Moore functor (8.2.3) arm(-,n): G=M"->Ab carries A to arm(A, n) = arm M(A, n) and carries ((p, qi): A --> B to arm(4p). Moreover there is the Moore bifunctor

(8.2.4) ar,;,"): GOP X G = (Mm)°P X M" - Ab which carries (A, B) to the homotopy group ar,;,")(A, B) = arm(A, M(B, n)) = [ M(A, m), M(B, n)]. 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA 259 Induced maps are defined as in (8.2.4). Since G and Ab are algebraic categories it should be possible to obtain purely algebraic descriptions of these Moore functors. For small values of m one has the following examples of such algebraic functors. For m = n the functor it"( -, n) carries A to A and (cp, rp) to gyp. For m = n + 1, n >_ 3, the functor 7t"+ (-,n) carries A to A ®1/2 and ((p, i/i) to cp ® 71/2. We get up to a canonical natural isomorphism the Moore functor Trn+2( -, n), n > 4, as follows.

(8.2.5) TheoremFor n > 4 the functor 7rn+2(-,n) carries A to G(A) and (cc, i/i) to iir, that is, there is an isomorphism irn+2(M(A, n)) - G(A) of groups which is natural in A (=- G.

Proof LetE irn + 2 M(71/2, n) = 71/4, n >- 4, be a generator. Then f induces an isomorphism

C*: G(A) = irn(Z/2, M(A, n)) = irrt+2M(A,n) (1)

which is natural in M(A, n) E M'. In fact for the pinch map q the composite (2) qC=71n+l: Sn+2-

is the Hopf map. This shows that the following diagram commutes

Ext(71/2, A ® 71/2) N Trn(71/2, M(A, n))- Hom(Z/2, A) (3)

A®71/2 7rn+2M(A,n) A*Z/2

Here the top row is the universal coefficient sequence and the bottom row is induced by the inclusion of the n-skeleton M(A, On c M(A, n); compare Proposition 6.11.2. The commutativity of the diagram shows that 6 * is an isomorphism, hencewe obtain the theorem by (8.2.1).

Next we consider the cohornotopy group

(8.2.6) -rrn+(A, S") = [M(A, n + 1), S" ],n > 3.

This group,as a functor in A E G, can be characterized asfollows.

(8.2.7) Lemma Forn >: 3 there is an isomorphism

0: irn+,(A,S") = Hom(G(A),Z/4)

which is natural in AE G, that is 0 o Hom(ir, 71/4) ° 6; see (8.2.2). 260 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA

Moreover the isomorphism 0 makes the following diagram with short exact rows commutative.

(8.2.8)

Ext(A,7L/2)) ° ' irrt+1(A,S") ' * Hom(A,Z/2)

Hom(A *71/2,72/4) H Hom(G(A),7L/4) -* Hom(A ® 72/2,72/4) The top row is the universal coefficient sequence and the bottom row is given by (8.1.2), hence this is a topological interpretation of the exact sequence in (8.1.2).

Proof of Lemma 8.2.7 We can derive the result from Theorem 1.6.4. Here we give an independent proof which defines the isomorphism 0 explicitly as follows. Since we are in the stable range we may assume n >_ 4. Let i): M(7L/2,n+1)-'S" be the Spanier-Whitehead dual of 6 in Theorem 8.2.5. Then the composite

17 rl,:SN+ 1 - M(71/2,n + 1) S", where i is the inclusion of the bottom sphere, is the Hopf map. This shows that r, induces an isomorphism U* Trn+ I(A, S) 17. [ M(A, n + 1), M(7L/2, n + 1)]

II Hom(G(A),G(71/2)) G(A,71/2) whereGUL/2) = 72/4.

Finally we consider examples of Moore bifunctors We clearly have binatural isomorphisms (A, B E G, n >_ 3) inn11(A,B)= Ext(A,B) (8.2.9) 1T,1"'(A, B) = G(A, B) where G(A, B) is the abelian group of morphisms A -* B in G. The next result yields an algebraic characterization of ir,,I+11, n >_ 4.

(8.2.10) TheoremLet 0G: G - SI'Ab' be the functor which carries B to the inclusion OG(B) = A: B ® 72/2 C G(B) and let n >_ 4. Then there is an isomorphism [M(A,n + 1), M(B,n)] _ B) =G(A,OG(B)) 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA 261 which is natural in A, B E G and which is compatible with A andµ in the universal coefficient sequence. Here G is the functor in Definition 8.1.3(B) and Addendum 8.1.3(C).

We point out that the natural isomorphism in Theorem 8.2.10 is available for all abelian groups A, B.

Proof of Theorem 8.2.10If A or B are finitely generated we obtain the following commutative diagram which is natural in M(A, n + 1) E M"'and M(B, n) E Mn.

'®'1 1 Ext(A,Z/2)®B it"+1(A,S")®B-0>Hom(A,71/2)®B

11 Ext(A,B(9 71/2) k

I'ls Ext(A, ir"+,M(B, n))N '1 » Hom(A,B®71/2)

Here k is given by composition, that is for a c 7r"..,(A, S') and be B = 7r,M(B, n) we have k(a ® b) = b a a. The top row, induced by (8.2.8), is exact though 0 ® 1 need not be injective. Moreover the bottom row is the universal coefficient sequence. Since the rows are exact the left-hand square of the diagram is a push-out diagram of abelian groups. Using (8.2.8) this push-out diagram corresponds exactly to Definition 8.1.3(B) with r)= AG(B). Hence weobtain the isomorphism in the theorem. If A and B are arbitrary abelian groups we use the following construction. For a space Y and the set B we have the one-point union

Y[ B ] = V YbwithYb = Y bEB which is a functor in Y and in B. The map -q: p,",- = M(Z/2, n + 1) -> S" in the proof of Lemma 8.2.7 induces the map

n[B] S"[B] ?I: M(71/2[BJ,n + 1)=p,'-[B] =M(71[B],n) which is natural in B. Moreover we have the inclusion

is M(Z[B],n) cM(B,n) 262 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA which is natural in M", that is ;pi = i(S"[cp1). We now consider the following diagram which is again natural in M(A, n + 1) and M(B, n). Hom(G(A),71/4[B])

II K(A, B) C Tr"+,(A, M(7Z/2[B], n + 1)) -AHom(A,Z/2[B])

71.1 push 1077). 1(n2). Ext(A,G(B)) Tr"+,(A,M(B,n)) Hom(A,B(&71/2) Here K(A, B) = kernel(p-,)1, µ is a functor in A, B E Ab and we identify G(B) = Tin+ 2M(B, n) by Theorem 8.2.5. The map i'q above induces (P2)* such that the diagram commutes. We observe that (P2)* p. can be identified with the map (P,) * A* in the diagram of Addendum 8.1.3(C). This yields the identification of bifunctors K(A, B) = Hom(A *11/2, K(B) *Z/2) via the inclusion j in Addendum 8.1.3(C). For the computation of -q# we first apply the natural map Q=A: ar"+,(A,M(B,n))=PRIN(dA,dB)->[d,,df®Z/2] to the diagram above; see Theorem 6.12.14 and (8.2.12) below. The second part in the proof of Lemma 6.12.13 shows that the composite K(A, B) Ext(A,G(B)) µ'-4 Ext(A, B *71/2) is trivial and hence iadmits a factorization Tj,: K(A,B)-_% A*Ext(A,B®71/2)=ker(µ*). It remains to show that for A = K(B)*7/2 the element %G) = 0(1) coincides with the homomorphism eB in Addendum 8.1.3(C). For this consider A =1/2 and y E K(Z/2, B) given by y E K(B) *71/2 as in Addendum 8.1.3(C). Then we have a subgroup j: B' c B generated by x; E B and we have y = j * y'. Now B(y') can be computed by Definition 8.1.3(B). This in fact shows that 6(1) = O. We have the commutative diagram

(8.2.11)

Ext(A,G(B))) .1 o 6(A, AM)µ -» Hom(A,B®71/2)

Ext(A,ar"+,M(B,n)) N Tr"+,(A,M(B,n)) - Hom(A,ir"+,M(B,n)) The left-hand side is the isomorphism given by Theorem 8.2.5. We now apply 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA 263 the push-out diagram (6.6.7Xii) to the bottom row of (8.2.11). This yields a connection of the Moore bifunctor 7T,,(+),with the Eilenberg-Mac Lane bifunctor 2, namely one has the binatural push-out diagram (A, B EG) Ext(A,G(B)) B) - Hom(A, B (9 71/2) (8.2.12) 1µ. push IQ 11 Ext(A, B *Z/2) >-+ B) -. Hom(A, B (9 71/2) Here the bottom is naturally split since,i= 0. Hence we can identify (n>-4) (8.2.13) H,,,+),(A, B) = Ext(A, B *71/2) ® Hom(A, B (9 71/2).

The operator Q coincides with Q in Corollary 6.6.8 andH,(, +',is the Eilenberg-Mac Lane functor which we already know to be split since H( 3 is split; see r = 2, m z 4 in (6.3.9).

8.3 The group rn+2 of an (n - 1)-connected space, n >_ 4 Whitehead's F-groups T X appear in the certain exact sequence. For a CW-complex X they are defined by the image (8.3.1) rmX=image( 7T Xm - t - TrmXm} where Xm is the m-skeleton of X. If X is (n - 1)-connected, then clearly I'm X = 0 for m < n. Whitehead computed the first non-vanishing group (8.3.2) n>_3. Here we do the next step and compute F,,, 2(X) for n > 4 Moreover we compute the F-group with coefficients (A, X) and we describe the functorial properties of these groups. The groups I' ,(X) and 1(A, X) depend only on the (n + 1)-type of the (n - 1)-connected space X. This (n + 1)-type is of the form (8.3.3) where K(71, n) is the quadratic space associated with the quadratic map

71 _(71,X: compare Definition 7.1.5. For n3 this is a stable quadratic map givenby a homomorphism is 71/2 -> The Postnikov map X -+K(ri,n) induces a natural isomorphism rF+ZX=r,,+,K(i,n) (8.3.4) r,+,(A, X) = rrt+i(A,K('h, n)). 264 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA

Hence as an abelian group rn+2(X) and rn+,(A, X) are determined by rr and (A, 71) respectively. The computation of the functor rn+2 is based on the following short exact sequence; see Theorem 5.3.7.

(8.3.5) PropositionLet n > 4 and let X be (n - 1)-connected. Then one has the natural short exact sequence

it1(X) ® Z1/2NF2(X) - Hn(X)*71/2 where 0(a ®1) = a'qn+ 1is induced by the Hopf map -%,1.

Proof We may assume that X is a CW-complex with dim(X) :!- n + 3 and X` -1 = *. Using Lemma (1.7.5) in Baues [CH] we obtain a map g: A -> B where A and B are both one-point unions of n-spheres and (n + 1)-spheres such that the mapping cone of g is homotopy equivalent to X. Moreover Hng is injective. Since we are in the stable range we thus have the following commutative diagram with exact rows.

g. ' 'rn+2B - In+2Cg Trn+1A -4 7Tn+1B U U U 1rn+2A - irn+2B -> rn+2X -> H,, (A) ®71/2 -> H,, (B) ®71/2 q, 92

The cokernel of g1 = g * is irn+ 1(X) ® 1/2 and the kernel of g2 = H,(g) ® 1 is H,,(X) * 71/2. This yields the required exact sequence which is natural since 0 is natural.

(83.6) CorollaryLet n >- 4 and let X be an (n - 1)-connected space with B = Hn(X). Moreover let f3: M(B, n) - X be a map such that H,,(,8) is the identity on B. Then one has the commutative diagram with short exact rows

'rn+l(X)®71/2 rn+2(X) H,(X)*71/2 I': T0. µ II B®71/2 N irn+,M(B,n) - B*71/2 where in carries b ®1 with b e B = an X to (brl,,) ®1. The bottom row is, on the one hand, the exact sequence in Theorem 8.2.5(3); on the other hand it is the exact sequence in Proposition 8.3.5 for X = M(B, n). We observe that the diagram in Corollary 8.3.6 is a push-out 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA 265

diagram of abelian groups which determinesthe group F., 2(X). The isomorphism C* in Theorem 8.2.5(1) has the followinggeneralization.

(83.7) TheoremLet X be an (n - 1)-connectedspace. Then there is a natural isomorphism* for which the following diagramcommutes

Ext(Z/2, X) H X) - µHom(71/2, 7r X )

11 11 "rn+,(X)®71/2 r,,+,(X)-_4

Here irn(Z/2, X) is the homotopygroup with coefficients in Z/2 and the top row is the universal coefficientsequence. The left- and the right-hand side denote the canonical identifications. Moreover,we obtain for 71 in (8.3.3) the isomorphism of groups

0: G(rt) = Irs+2(X)

which is compatible with A andI.L.Here G(TI) is the group in Definition 8.1.3(A).

Proof The isomorphism carries a: M(71/2, n) _C14 X"' C X to the element of 2X given by the composite _, Sn+2 , M(71/2, n)

As in the proof of Theorem 8.2.5we see that the diagram in Theorem 8.3.7 commutes. Moreover we define 0 by Oi= A and 0i =13 ** where ;r * is the isomorphism in Theorem 8.2.5(3)and where we use f3,,in Corollary 8.3.6. Compare also Theorem 1.6.11 where G(71/2,-q) = G(rt).

(83.8) Remark By putting X= K(B, n) in Corollary 8.3.6 we readily get, for n>4, Hi+3K(B,n)=f,,,2K(B,n)=B*71/2 and the operator

Q: n) = G(B) - Hi+3K(B, n) = B *71/2 in Theorem 6.6.6 is surjectiveand coincides with µ in Corollary 8.3.6. We use the next result for the computation of F,,. (A, X)- Here 0 in Corollary 8.3.6 inducesan isomorphism r,,I( /3 ). 266 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA

(8.3.9) Lemma Let X and r8 be given as in Corollary 8.3.6. Then one has the commutative diagram

Ext(A,rn+2X) 11 ' rn+1(A,X)µ *

Ext(A,0.) push 0.

Ext(A,ir,,,.2M(B,n)) "'I 7rn+1(A,M(B,n)) -. Hom(A,B®7L/2)

The top row is the universal coefficient sequence (Definition 2.2.3) which is natural in X. The bottom row is the universal coefficient for X = M(B, n) which was algebraically characterized in Theorem 8.2.10. Since the diagram in Lemma 8.3.9 is a push-out diagram we obtain by composing this diagram and diagram (8.2.11) the isomorphism

(8.3.10) G(A,A(-n® 1))-I'i+1(A,X).

Here we use the composite

,1®11 irn(X) ®1/2 irn+1(X) ®1/2° G(71).

The left-hand side of (8.3.10) is defined by the bifunctor G in Definition 8.1.3. By putting X = K(B, n) in (8.3.10) one has rl = 0 and hence one has the binatural isomorphism

Hn+,(A, K(B,n)) = I'n+1(A, K(B,n)) = Ext(A, B * Z/2) ® Hom(A, B ®1 /2).

Here the second equation is a consequence of (8.3.10). We discussed this already in (8.2.13). We now study the functorial properties of Theorem 8.3.7 and (8.3.10). For this we need the operator f - 6,, given as follows.

(8.3.11) DefinitionLet A, B, R be abelian groups. Then one has a homomorphism R),6-6#, which is natural in A and B. If A I -A0 -A is a short free resolution of A then 6 E Ext(A, B) is represented by 1 E Hom(A1, B) and 1:,,is the composite

,1, : A * R cA 1 ®R -- 11 B 0 R. 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA 267

Let B > E -.A be an extension of abelian groups representing i;. Then there is the classical six-term exact sequence 0->B* R-->E*R--)-A*R4 B®R->E®R-A®R-*0

where 6# is the boundary operator. Using the inclusion 0: Ext(B,ir , , X) >--> ir (B, X) we define for /3 E 7rn(B, X) and r` E Ext(B, ir +, X) the difference homomorphism (/3+Ai:)* where /3* is the same as in Corollary 8.3.6. On the other hand we have (/3+A6)* -/3*: where /3* is the same as in Lemma 8.3.9.

(8.3.12) PropositionThe homomorphism (*) coincides with the composite

f* ir , , +2M(B,n)-.B *7L/2 ®71/2NF,,,(X). Moreover (* *) coincides with the composite it JA, M(B, n)) Q1 Ext(A, B *7L/2) 17,,(A,X) where Q, is the first coordinate of Q in (8.2.12); see (8.2.13).

We leave the proof as an exercise; compare the more sophisticated unstable version of Proposition 8.3.12 in Theorem 11.4.7 below.

8.4 Proof of the classification theorem 8.1.6 We apply the classification theorem 3.4.4 where we setr = 2 and C = types;, = I'G, n >- 4; compare 7.2.9. Since spaces,3, = spaces,3,(C) we obtain the detecting functor

A': spaces -- Bypes(FG, F) (1) where F is the following bype functor on FG. For r): Ho ® Z/2 - 7r,let F,(i7) = G(,1) (1) as in (8.1.3) and F0(rl) = kernel(rl: Ho (9 71/2 - ir,) (3) 268 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA

Moreover we define the bype functor_F on 1'G by the following pull-back diagram where we use the functor G in (8.3.10) and where we use the inclusion j: Fo(rd) c Ho ® Z/2.

Ext(A,F(i7)) 6(A,O(r7 (& 1))-µ, Hom(A, Ho (9 Z12)

pull Ji. (4) II F(A,-q) `0 Hom(A,Fo,7) Now Lemma 8.3.9 and the classification theorem 3.4.4 yield the detecting functor A' in (1). We claim that we have a forgetful functor 0: Bypes(l'G, F) -> A3-System (5) which is also a detecting functor. This yields the result in Theorem 8.1.6 by use of the detecting functor ¢A'. The functor ¢ is essentially the identity on objects; compare for this the definition of F-bypes in Section 3.2 and the definition of A3-systems in Definition 8.1.4. Let (P;: H; -> H; be homomorphisms. Then 0 carries the morphism (X, (P2, 453) in Bypes(FG, F) with X = {(P,r, , coo,'Po}: -q - 77' in 1'G to the morphism (P = ((POI (P21 (P3, (P,,,(Pr) in A3-System with cPr = X* : F = F'= G(,q'). (6) We have only to check that ¢ is a full functor, that is, for (Pr there always exists X such that (Pr = * as in (6). For this we consider the diagram

G(i) G(H0) ® ir, ® 1/2

wrI o1(450. P- ). (7)

G07')G 0y 7 ' )' )-- -' G(Ho) ®ir', ®71/2 where (cpo, (P,,, ) * has the coordinates ;PO: G(H0) -, G'(Ho), 7r,®71/2->7r,®71/2, µ: G(H0) Ho *l/2 -> rr, ® 71/2. The definition in (8.3.11) shows that the right-hand side of (7) induces X* in (6). We claim that for 'Pr there is (;po, (P,,, ) such that diagram (7) commutes. To see this we first choose a map ;PO: G(H0) - G(Ho) compatible with (Po. Then ;Po ®(P ® 1 induces a map (Pr: G(q) -* G(71') compatible with 0 and A. Since also (Pr is compatible with A and µ in Definition 8.1.4(8) there is 5 E Hom(Ho *l/2, rrl ® 71/2) 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA 269

with cpr = (pr + ASµ. Since the operation H # is given bythe surjection

Ext(H0, iri) Q-- Ext(Ho, ir, (& 71/2)= Hom(H0 *71/2, ir' (9 71/2)

we see that there exists with = S. This implies that diagram (7) commutes for Therefore 4 is a full functor and hence a detecting functor. Similarly we get the detecting functor A' in Theorem 8.1.6 byuse of A' in Theorem 3.4.4.

8.5 Adem operations

Algebraic data which classify homotopy typesare by no means well determined or unique. Itis, however, suitable to search for such data which directly refer to basic invariants like homology groups and homotopygroups. For example, the A3-systems in Section 8.1 which classify (n- 1)-connected (n + 3)-dimensional polyhedra Xuse the homology groups H* X of X with 71-coefficients, and the homotopygroups 7r" + i X, 'rn12X can easily be computed by the A3-system associated with X. On the other hand, there is a classical approach to classifying homotopy types X by cohomology groups with coefficients in various abelian groups and by cohomology operations. It is a kind of old belief that at least in the stable range itis possible to find such cohomological data which classify homotopy types. Theuse of cohomology requires the restriction to spaces with finitely generated homologygroups. Given such cohomological data it is then a hard problem to determine the homotopy groupsirn X for n < dim X in terms of these data;see for example (5.3.5). Actually only a very few complete results are known on the classification of homotopy types via cohomology and cohomology operations. J.H.C. Whitehead [HT] for example used Steenrod squares for the classification of (n - 1)-connected (n + 2)-dimensional polyhedra,n > 3.Such Steenrod squares are primary cohomology operations. It is not possible to classify (n - 1)-connected (n + 3)-dimensional polyhedra only by primary cohomology operations. In fact for the double Hopf map 77,1: S"+' S" we have the mapping cone

X= Sn U,72en+3 and all primary cohomology operationson X are trivial. It is a classical result of Adem that there isa non-trivial secondary cohomology operation forX showing that -qzis essential. This suggested the classification of (n - 1)- connected (n + 3)-dimensional homotopy types, n > 4, by primary and secondary cohomology operations;compare the papers of Shiraiwa, Chang, and Chow. The results turnedout to be very intricate and unclear. Here we 270 8 (n- 1)-CONNECTED (n+ 3)-DIMENSIONAL POLYHEDRA follow the work of my student S. Jaschke [AO} who shows that indeed a classification via primary and secondary cohomology operations is possible. For this one needs the notion of an A3-cohomology system which is a modification of the concept of Chang and which relies on secondary cohomology operations of Adem type. (8.5.1) Notation We describe some fundamental concepts of homotopy theory, concerning extensions, coextensions, lifts, and colifts, respectively. Consider maps

A14B 9Y withgf = 0inTop*. (1)

Given the null homotopy H: gf = 0 we obtain the following maps depending on H

Cf -L Y,extension of g, (2)

E ALCg,coextension of f, (3)

A f Pg,lift off, (4)

Pf - fIY,colift of g. (5)

Here Gg = Y Ug CB is the mapping cone given by the push-out

119 CB Cg

101 push Ti8 (6) B -- Y 9 where CB = I X B/I X * U (1) x X is the cone on X. On the other hand, Pg = B Xg WY is the fibre of g given by the pull-back

(7)

where WY is the contractible path object of Y, that is, WY= {oY', Q(1) = * } and go(o,) = o-(0). A null homotopy H: gf = 0 as above can be identified with maps

H: CA Y andH: A - WY (8) 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA 271 respectively. The extension g is the map with gig = g and girg = H; the lift f is the map with qgf =f and 7rg f = H. Moreover for the suspension 1A and the loop space fly we obtain the coextension f and the colift g above follows. Consider the commutative diagrams

T I T A --> B -- Y (9)

1 H CA w(g) WY

I I I A- B --- p Y (10)

jY These diagrams yield the maps

(irgC(f),igH) f: IA=CAuACACg (11) (W(g)lrf, H ) g: Pf WY x,, WY= fY. (12)

Here the homeomorphisms are chosen such that

If (13)

fig=gio: flBcPf-->fly. (14)

The map q0 is the pinch map Cq --, Cg/Y= EB and io is the inclusion of the fibre SZ B = (qg)-' ( *) c Pf. More generally extension and coextension can be defined in any cofibration category and lift and colift are the strictly dual constructions in a fibration category; see Baues [AH].

The Steenrod operations are homomorphisms

(8.5.2) Sq": H"(X,71/2) +H"+"(X,71/2) which are natural in X. They determine up to homotopy maps

Sq": K(71/2,k)- K(71/2,k+n) (1) 272 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA which induce (8.5.2) via the isomorphism Hm(X, ar) = [X, K(zr, m)]. The operator Sq' coincides with the Bockstein homomorphism associated with the 2 4 exact sequence 71/2 1/4--1/2 so that

2 4 H"(X,71/2)µ°_H"(X,71/4) AZ-H"(X,7L/2)-H"+'(X,71/2)-µ-°

(2) is a long exact sequence; in particular p4Sq' = 0 and A2Sq' = 0. Hence also Sq'µ°2 = 0 where µz: 71-> 71/2 is the quotient map. The Adem relations

= Sq'Sg2 (8.5.3) Sq3Sg3Sq' + Sq2Sq2 = 0 give rise to the following composites gf which are null homotopic (compare for example Mosher and Tangora [CO]).

(Sq',Sg2) (Sg3,Sg2) K(71/2, n) - K(71/2, n + 1) x K(71/2, n + 2) - K(71/2, n + 4) (1)

(Sq'Sq',Sg2) K(71/2, n) K(7112, n + 3) x K(71/2, n + 2) (sg'.sg2) K(71/2, n + 4)(2) Sg3'2- sqZ K(71, n) K(71/2, n + 2) - K(71/2, n + 4) (3)

sq2 µ4sg2 K(71/2, n)---> K(7/2, n + 2) K(71/4, n + 4) (4) Sg sq2 K(71/4, n) 2)K(71/2, n + 2) 0 K(71/2, n + 4). (5) Given a composite gf: A -> B -> Y, gf = 0, as in (1)-(5) we choose a colift of g as in (8.5.1) and we define the secondary operation 0 associated with gf = 0 as follows: Consider the diagram s Pf fly X1f X A -a B- Y where fu = 0. Then we choose a lift u of u and gu represents cb(u). Hence given {u} E [ X, A] with f,, {u} = 0 we obtain the well-defined subset 4i{u} C [X, flY] consisting of all composites gu where g and u are given by homotopies 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA 273 fu = 0 and gf = 0 respectively. For the maps in (1)-(5) above this subset is actually a coset of a subgroup of [X,11Y]. This way one derives from (1)-(5) the following well-defined Adem operations (1)'-(5)' Hn+3(X,71/2) H"(X,71/2) z) ker(Sq') n ker(Sgz) --> (1) im(Sg3 +Sqz)

Hn+3(X,71/2) H"(X,71/2) D ker(Sq-) n ker(SgzSq') ---im(Sg' + Sq 2) (2) Hn+3(X,71/2) 62o H"(X,71)ker(Sq µ°) - (3) im(Sgz) Hn+3(X,71/4) 6,2 H"(X,71/2) 3 ker(Sgz) zS (4) im( 4z) Hn+3(X,71/2) asZ H"(X,71/4) D ker(Sgzµ2) (5)' im(Sgz) where im(Sq' + Sqs) = im(Sq') + im(Sgs) is the sum of subgroups. Originally Adem used `chain maps' for the definition of such operations. In (3) we also write Sgzµ° = Sq2 and 0° coincides with the operation used in Theorem 5.3.8(b). One can check that the following diagrams commute, where the arrows i and q denote inclusions and quotient maps respectively 0° ker(Sq 2AO ) H"+3(X,71/2)/im(Sg2)

µz iq (8.5.4) ker(Sq z) n ker(Sq' ) H"+3(X,71/2)/im(Sg3+ Sqz )

1t 1q ker(Sgz) n ker(Sg2Sq') 46* Hn+3(X,71/2)/im(Sq' +Sqz)

ker(Sq 2A4 H" +3(X, 2)

µz lq (8.5.5)ker(Sgz) n ker(Sq') H"+3(X,71/2)/im(Sg3 +Sqz)

µs z m Hn+3(X,71/4)/im(µa Sq2)

We are now ready to introduce the notion of an A3-cohomology system which describes further properties of the Adem operations 04 and0z. 274 8 (n - 1)-CONNECTED (n + 3)-DIMENSIONAL POLYHEDRA

(8.5.6) Definition An A3-cohomology system is a tuple

S=(H*(0), H*(2), H*(4), 0(2), 0(4),µ(2),µ(4),µ4> Sqo, 592, 04>02 ) with the following properties. (a) H*(O), H*(2), H*(4) are graded finitely generated abelian groups concentrated in degree 0, 1, 2, 3, and H°(0) is free abelian. (b) µ4: H*(2) - H*(4) and µ2: H*(4) -H*(2) are homomorphisms of degree 0. (c) A(2), z(4) and µ(2), µ(4) are homorphisms of degree 0 and 1 respectively for which the following diagram commutes H*(0) ®7L/2;o(2))H*(2) µM H*(0) *7L/2

11®µy

H*(0)®71/4-> H*(4)µc4)_ H*(0)*71/4

l+2 1µz µ2

µc2) H*(0) ® 71/2 > H*(2) 4 H*(0) *7L/2 The rows are split short exact sequences and splittings can be chosen which extend the diagram commutatively, hence µ2 µ4 = 0. (d) Sqo: H *(2) - H *(0) and Sq 2: H *(2) - H *(2) are homomorphisms of degree 1 and 2 respectively. (e) For r= 2,4 we define µ°: H*(0) -> H*(r) by µ,°(x) = o(-rXx ® 1) and we set Sq' = Ao Sqo With this notation the following diagram commutes and has exact rows.

s`'° H'(0) 2 H'(0) µ2 H'(2) . H'+'(0) 2

I A02 I I-01 I µi

H'(2) µ` H'(4)µz * H'(2) Sq H'+'(2) µ

(f) H°(2) Dker(Sg2) --H3(4)/im( µ4Sg2) mz H°(4) D ker(Sg2µ2) - H3(2)/im(Sg2) 8 (n - 1)-CONNECTED (n+ 3)-DIMENSIONAL POLYHEDRA 275

are homomorphisms of degree 3 such that the following three diagrams commute.

ker(Sg2µ2) z.H3(2)/I im(Sg2)

IF,4 04 ker (Sq2)- H3(4)/im( µ4Sg2)

m< ker(Sg2) H3 (4)/im( µ4Sg2) lii H3(2) sq2 H°(2) H2(2) µ4 jsq' ker(Sg2µ2 b H3(2)/im(Sg2)

A morphism f: S -* S' between A3-cohomology systems is given by a tuple of homomorphisms of degree 0

f(0): H*(0) 3H*(0)' f(2): H*(2) -4H*(2)' f(4): H*(4) _H*(4)' such that f is compatible with all operators and diagrams above. Let A3-cohomology be the corresponding category. This is an additive category with the obvious notion of direct sum of A3-cohomology systems given by the direct sum of abelian groups.

Recall that A3 is the full homotopy category of (n - 1)-connected (n + 3)- dimensional polyhedra which are finite.

(8.5.7) TheoremLet n >: 4. One has a detecting functor

A: AR -A 3-cohomology which carries a space X to the A3-cohomology system A(X) given by H'(T) = H"+`(X,7L/T) for r= 0, 2, 4 and i E Z. Moreover A(T) and µ(T) are operators of the universal coefficient sequence and µ2, µ4, Sqo', Sq2 are defined in (8.5.2) 276 8 (n- 1)-CONNECTED (n+3)-DIMENSIONAL POLYHEDRA and 04, ¢z are the Adem operations in (8.5.3)(4)', (5)'. The functor A is an additive functor.

Hence isomorphism types of A3-cohomology systems are in 1-1 correspondence with homology types of finite (n - 1)-connected (n + 3)-dimensional polyhedra, n >_ 4. The theorem was essentially obtained by Chang. Jaschke [AO] wrote down a complete proof. It shows that the classification via cohomology operations is considerably more complicated than the classification via boundary invariants in Section 8.1. The examples in (10.2.16) show that the pair of operations 464, 04 is needed for the classification. The classical Adem operations q5', 0", 0z do not suffice to detect all possible homotopy types in A;,. 9 ON THE HOMOTOPY CLASSIFICATION OF 2-CONNECTED 6-DIMENSIONAL POLYHEDRA

In this chapter we describe algebraic models which characterize the homotopy types of 2-connected 6-dimensional polyhedra. Such polyhedra are in the metastable range so that diverse features of `quadratic algebra' are involved in the classification. We proceed in a similar way as in Chapter 8 where we classified (n - 1)-connected (n + 3)-dimensional homotopy types which are in the stable range n >_ 4. We apply boundary invariants which, via the classification theorem 3.4.4, yield algebraic classifying data for 2-connected 6- dimensional homotopy types.

9.1 Algebraic models of 2-connected 6-dimensional homotopy types

We introduce the purely algebraic category of A3-systems. The main result of this chapter shows that A3 -systems are algebraic models which classify 2-connected 6-dimensional homotopy types. Recall that we have the exterior square which is the functor (9.1.1) A2:Ab-->Ab defined by A2(A) =A ®A/(a ® a - 0). Let H and L be abelian groups. A quadratic A-map is a homomorphism A:H®7L/2ED A2(H)-*L which admits a factorization A: H®7L/2ED A2(H) Here we use (9.1.2) below. Let AAb be the category of such maps; objects are quadratic A-maps and morphisms (qi, alro): A -b A'are pairs(i/rl, 4,0) E Hom(L, L') ® Hom(H, H') for which the diagram c4'°> H®7L/29A2(H) H'®7L/2®A2(H') IA, IA L 278 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA commutes. Here we set ('Iio)* =1#® ® 71/2 ® A2(00). We have for any abelian group A the extensions

(9.1.2) A ® 71/2HG(A) - A *1/2

(9.1.3) Ext(A,l/2) N Hom(G(A),71/4) Hom(A,71/2) as in (8.1.1) and (8.1.2). We use (9.1.2) for the definition of morphisms in the category G; see Section 1.6. Recall that QM(71) denotes the category of quadratic 71-modules; see Section 6.13. We introduce a functor

(9.1.4) A,: GOP -, GM(71) as follows. Let d: Hom (A, 71/2) -> Ext(A,71) be the connecting homomorphism induced by the exact sequence Z 71 - 71/2. Then we define for an object A in G

A1(A) _ (Hom(G(A),71/4)dµExt(A,71) -L Hom(G(A),71/4)) where H = dµ is given by µ and d above and where P = 0 is trivial. The functor A, carries a morphism (cp, ip): A ---> B in G to the map ((p, ilp)*: A1(A) -* A,(B) in QM(71) given by Hom (;p,Z/4) and Ext We observe that the exact sequence (9.1.3) induces the following short exact sequence of quadratic 71-modules

(9.1.5) A0(A) >- A,(A) -µ Hom(A,71/2) which is natural in A E G. Here we set

A0(A) _ (Ext(A,71/2) -o- Ext(A,71) -s--i Ext(A,l/2))

1° 1° 11 1° A1(A) _ (Hom(G(A),71/4)H- Ext(A,Z) - Hom(G(A),71/4))

Hom(A,71/2) _(Hom(A,71/2) -i 0 Hom(A,71/2))

The quadratic 71-module corresponding to an abelian group D is given by D = (D --> 0 -* D). Hence one has the isomorphism of quadratic 71-modules.

(9.1.6) A0(A) = Ext(A,71/2) ® 71^ ® Ext(A,71) 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA 279 where 7L° = (0 --> Z -* 0) is the quadratic 7L-module associated with the exterior square functor A2 in (9.1.1). We now apply the quadratic tensor product which is a functor

®: Ab x GM(7L) -* Ab.

Then (9.1.6) shows that for an abelian group B we have the isomorphism in the top row of the following diagram

(9.1.7) B ®AQ(A) = B ®Ext(A,LL/2) ®A2(B) ®Ext(A,LL)

C2 16 161 ® Ext(A,B®7L/2®A2B)= Ext(A, B ® Z/2) ® Ext(A, A2B)

Here e,, e2 on the right-hand side are the evaluation maps with e,(x ®y) = (x®7L/2)*(y) and e2(u®v)=u*(v) where x: 7L-*B and u: 7L-->A2(B) denote the homomorphisms given by x E B and u E A2(B) respectively. We are now ready for the definition of the bifunctor G which replaces the bifunctor G in Definition 8.1.3.

(9.1.8) (A) Definition We define a functor

G: GOP X AAb - Ab as follows. If A or H are finitely generated let G(A, A) with A as in (9.1.1) be given by the push-out diagram with exact rows:

Ext(A, L) G(A, A) - Hom(A, H ®7L/2) - 0

A* T push Ext(A, H ®7L/2 ®A2H) )i. 11 1 He. H®A0(A) -H®A(A)H° ® H ®Hom(A,7L/2) - 0

Here the bottom row is obtained by applying the quadratic tensor product He to the short exact sequence (9.1.5). The bottom row need not be short exact. For a morphism ((p, ipp): A -+ B in G we obtain

(,P,;P)':G(B,A),d(A,A)

H®A,('p,) 280 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA and for a morphism (1/rl, 1/ro): A -A' in AAb as in (9.1.1) we get

(+61, 4io) * : A1(A, A), A1(A, A') (iri,iro)* =Ext(A,r/i1)®,/ro®G(A).

(9.1.8) (B) DefinitionIn general we have to use the following intricate definition of the group G(A, A) which canonically coincides with Definition (9.1.8) (A) if A or H are finitely generated. Here we use notation as in Addendum 8.1.3 (C). The short exact sequence of quadratic 71-modules in (9.1.5) yields for A = Z/2 the following commutative diagram

K(H) >-* 71/4[H] P2P 1B" jo cok(TH) ®A2HHH® A1(Z/2) -+H®1/2 in which the bottom row is short exact. Here the homomorphism 0' carries [x] with x e H to 0'[x] =x ® 1 where 1 E 1/4. For this recall that A1(71/2) _ (1/4 -L Z/2 -° Z/4). Let

6H: K(H)*71/2 -* cok(TH) ® A2H be the restriction of O. One can check that BH = (OH, yHp) is given by 8H in Addendum 8.1.3 (C) and by yH in Definition 6.2.13(B). We now obtain G(A, A) by the following push-out diagram

Hom(A *7L/2, K(H)*7L/2) H Hom(G(A),71/4[H])

0*Ext(A, H®71/2 ® A2H) push

A.I Ext(A, L) G(A,A) µ » Hom(A,H®71/2) Compare the diagram in Addendem 8.1.3 (C). The natural transformation 0 is defined by 0(a)=a*OH with BH E A * Ext(K, H ® 1/2 ® A2H) = Hom(K, Cok(TH) ® A2H ) 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA 281 where K = K(H) * 7L/2. This completes the definition of the bifunctor G in Definition 9.1.8 (A). Using the notation on the group G(,7) in Definition 8.1.3 and the group G(A, A) in Definition 9.1.8 above we are now ready to define algebraic models of 2-connected 6-dimensional homotopy types which we call A 3 systems.

(9.1.9) Definition An A3-system

(1) is a tuple consisting of abelian groups H3, H5, H6, Tr4 and elements

b5 E Hom(H5, H3 (9 71/2)

E Hom(H3 ® 71/2, ir4 )

b6 E Hom(H6,G(rl) ® A2H3)

E G(H5,71, ). (2)

Here X10 = q(A(n (9 1) ® AzH3) is the composite

770:H3®1/2®AZH3-->G(77)®AzH3- cok(b6) (3) where q is the quotient map for the cokernel of b6 and where the first arrow, A(r, ® 1) e AZH3,isinduced by A(i (9 1): H3 ® 71/2 - vr4 ® Z/2 --b G(,1); compare the definition of G(71) in Definition 8.1.3. These elements satisfy the following conditions (4) and (5). The sequence

H5 b, H3 ®7L/2 17 > 1r4 (4) is exact and /3 satisfies

A(/3)=b5 (5) where µ is the operator in Definition 9.1.8. A morphism

(6) between A3-systems is a tuple of homomorphisms

,p;: H, H, (i = 3,5,6) Pn'T4 - 74 cpr: G(7) --> G(rt') 282 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA such that the following diagrams (7), (8), and (9) commute and such that equation (10) holds.

H5 b, H3 ® 71/2 IT4

l'Ps 1Wn (7) H5'bH3 ® 71/2 1r4

7r4®71/2 G(i1) --->H3*Z/2

(8) wn®1 I(P r I w3.1 1r4' ® 71/4NG(ri') - H3 *Z/2

H6 - GOO ®A2H3

1'P6 I(Pre A2('P3) (9) H6bG(l') ®AZH3

Hence'Pr ® A2(!p3) induces(Pr ® A2(4)3):cok(b6) - cok(b6) suchthat A2(43)): 771 - rla is a morphism in AAb which induces (P31 'Pr (B A2((p3))* as in Definition 9.1.8. We have

('P3,4PrED A2((P3))*(/3)=(43,43)*0') (10) in G(H3, rro ). Here we choose irp3 for cp3 as in (8.1.1). The right-hand side of (10) does not depend on the choice of ;P3; compare Definition 9.1.8. An A3-system S as above is free if H6 is free abelian, and S is injective if b6: H6 - I'(-q) ® A2(H3) is injective. Let A3-System, resp. A3-system, be the full category of free, resp. injective, A3-systems. We have the canonical forgetful functor

0: A 33-System -* A 3-system3 (11) which replaces b6 by the inclusion b6(H6) C G(q) ® A2(H3) of the image of b6. One readily checks that the forgetful functor 0 is full and representative.

(9.1.10) Definition We associate with an A3-system S as in Definition 9.1.9 the exact r-sequence H5b,H3®71/217ir4- H6-G(7I)®A2(H3)-.IT5-s 114->0. Here H4 = cok(q) and the extension

cok(b6) - IT5 -. ker(bs) (1) 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA 283 is obtained by the element /3 in Definition 9.1.9, that is the group -rrs= ir( /3t) is given by the extension element /3t E Ext(ker (b5), cok(b6)) defined by

of = A-t (j, j)"( /3). (2)

Here j: ker(bs) c H5 is the inclusion. The element /3t does not depend on the choice of (j, j) in G. Compare (2.6.7).

Recall that spaces3 is the full homotopy category of 2-connected 6- dimensional CW-spaces X and that types; is the full homotopy category of 2-connected 5-types. We have the Postnikov functor P: spaces3 , types3 which carries X to its 5-type.

(9.1.11) TheoremThere are detecting functors

A': spaces33 --> A3-Systems3 A': rypes3 -A3-systems.

Moreover there is a natural isomorphism 4A'(X) = A'P(X) for the forgetful functor 46 in Definition 9.1.9 (11) and for the Postnikov functor P above.

Let S be a free A3-system. The detecting functor A' in Theorem 9.1.11 shows that there is a unique 2-connected 6-dimensional homotopy type X = Xs with A'(X) = S. Then the r -sequence for S is the top row of the following commutative diagram

(9.1.12)

H6 --,G(el)®AZ(H3)-.Tr(St)- H5 ->H3®7L/2 '7 7r4 - H4

II II =1 II II II II H6X f5X 7r5X-'H5X-=F4X ' 1r4X-.H4X

The bottom row is Whitehead's certain exact sequence for X. The diagram describes a weak natural isomorphism of exact sequences.

Proof of Theorem 9.1.11Using similar arguments as in Section 8.4 one gets the theorem by the results in the following sections 9.2 and 9.3. In particular one has the canonical forgetful functor

0: Bypes(FG, F) ->A3-Systems 284 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA which is a detecting functor so that A' in the statement of the theorem is the composite of 46 and the detecting functor A' in the classification theorem 3.4.4.

RemarkIn Theorem 6.4.1 we describe the detecting functor

A': spaces(3,5),r Bypes(Ab, H53)) (1) where H53> is the Eilenberg-Mac Lane functor which we compute in 9.2.11 below and which satisfies

H53'(A, B)= G(A, A). (2)

Here A: B ®Z/2 ®A2B - B * Z/2 ®A2B is given by A = 0 E ) see Re- mark 9.2.12. One readily checks that the functor A' in Theorem 9.1.11 actually yields a commutative diagram

n-,, spaces; A3-Systems U U spaces(3, 5) Bypes(Ab,

In fact, bypes in Bypes(Ab, H53)) can be identified via (2) with free A3- systems for which ir4 = 0. In Theorem 6.4.1 we also describe the detecting functor

A: spaces(3,5),, Kypes(Ab, H(s)) by use of k-invariants. The corresponding detecting functor A for spaces3 which also uses k-invariants, however, is not known.

(9.1.13) Definition We define the suspension functor

Y.: A 3-Systems --> A 3-Systems (1) as follows. First we obtain the natural map I between quadratic Z-modules given by

A,(A) = (Hom(G(A),l/4)dµEWt(A,7) -°+ Hom(G(A),Z/4)) II £1 1I I Hom(G(A),//4) = (Hom(G(A),7L/4) 0 - Hom(G(A),Z/4)) 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA 285

Now let S be an A3-system as in Definition 9.1.9. Then we have for iEJ the commutative diagram

-qa:H30 71/2(D AZH3->G(i7)a) A`H3 cok(b6) I P I IPI l- 71#: H3 (9 71/2 --> G(q) cok(p,b6) where p, is the projection and where i1# = g061(9 1) as in Definition 8.1.4(3). Using I and if above we obtain the homomorphism

E: G(H3, rlo) --> G(H3, 17*)

I=Ext(H3,o,)®H3®1. (2) For this compare the definition of the bifunctors G and G respectively. The suspension functor I above is now defined by XS) = S' where S' is the A3-system given by

(3)

Moreover the suspension functor is the identity on morphisms.

We now obtain for n >_ 4 the following diagram of functors

spaces3 spaces,3,

(9.1.14) A1 IA' A3 Systems A3-Systems

Here E" - 3is the (n - 3)-fold suspension functor for spaces and is the algebraic suspension functor in Definition 9.1.13. Moreover A' denotes the detecting functor in Theorems 9.1.11 and 8.1.6 respectively.

(9.1.15) TheoremDiagram (9.1.14) commutes up to a natural isomorphism, that is, for a 2-connected 6-dimensional polyhedron X one has an isomorphism of A3-systems A'(1n-3X) =I(A'X) which is natural in X, n >- 4.

Since by the Freudenthal suspension theorem the functor I" -3 isrepre- sentative, also the algebraic functor I is representative; this also can be readily seen by the definition of I. As an application we get 286 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA

(9.1.16) CorollaryLet X be a 2-connected 6-dimensional polyhedron with H3X cyclic. Then the homotopy type of X is determined by the suspension EX, that is IX= Y.Y implies X= Y.

Proof Since H3 X is cyclic we have A2H3 X = 0. Hence p, b6 = b6 in S' of Definition 9.1.13 (3). Moreover I in Definition 9.1.13 (2) is an isomorphism since I is compatible with A and A.

Similarly we derive from (9.1.12) the

(9.1.17) CorollaryLet X be a 2-connected polyhedron with H3 X cyclic. Then the suspension Y.: vr5 X = ir6I X is an isomorphism.

For example we get by Proposition 8.1.10

(9.1.18) ir5(y(y.2R P)= Z/8.

9.2 On ir5 M(A, 3)

We compute the homotopy groups ir5M(A,3) and 7r4(A,M(B,3)) of Moore spaces and we determine the functorial properties of these groups. As special cases of (8.2.3) we consider the functors

(9.2.1) TT5(-,3): G = M3 -' Ab

(9.2.2) ir43): GOP x G = (M4)OP X M3 - Ab which carry A to 7r5M(A,3) and (A, B) to ir4(A,M(B,3)) respectively. We want to describe the functors above up to a canonical natural isomorphism by purely algebraic functors defined via the algebraic structure of the category G.

(9.2.3) Lemma One has a natural short exact sequence

A®1l/2®AZ(A) >* 7r5M(A,3) -.A*l/2.

Moreover the suspension Y yields the following short exact sequence which is naturally split.

A2(A) ° ir5M(A,3) - ir6M(A,4)

Proof We apply the second sequence in Corollary 6.15.15 for n = 3. Since ir5{S") = 1/2 9 l^ we get A 0 ir5(S") =A 0 11/2 9 AZ(A). Moreover 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA 287

Ir4(S") = 1/2 so that the first sequence of the lemma is a special case of Corollary 6.15.15. Next the Freudenthal suspension theorem shows that the second sequence is short exact. Moreover this sequence is naturally split since the composite I ir4 M(A, 2) c Ir5 M(A, 3) -i a6 M(A, 4) is an isomorphism. The lemma yields by Theorem 8.2.5 the following result.

(9.2.4) TheoremThe functor ir5(-,3): G -* Ab carries A to G(A) ® A2(A) and carries (gyp, l') to® A2(cp ), that is, there is an isomorphism 7r5(M(A, 3)) _ G(A) ® A2(A) which is natural in A E G.

This is the purely algebraic description of the functor 7T5(-,3) in (9.2.1). Let Add(Z) be the category of finitely generated free abelian groups. We consider the quadratic functor

(9.2.5) 7r4: Add(Z) -> Ab which carries B to the homotopy group with coefficients A, vr4(A, M(B, 3)). This functor is determined by the quadratic 11-module

Trq {S3} _ (7r4(A, S3) 7r4 (A, SS) --> 7r4(A, S')) (1) as in Definition 6.13.10 (4). Here we have

ir4(A,S5) = Ext(A,71) (2)

ir4(A,S3) = Hom(G(A),71/4). (3)

Both isomorphisms are natural in A E G; compare Lemma 8.2.7. For the quadratic 71-module A,(A) in (9.1.4) we obtain the following topological interpretation:

(9.2.6) PropositionThere is an isomorphism vr4 (S3} = A1(A) of quadratic 71-modules which is natural in A E G

Proof Using the isomorphism rl*in the proof of Lemma 8.2.7 we see that each element x E ir4(A, S3) is a composite x: M(A,4)

yis a suspended map. Hence H in (9.2.5) (1) is determined by the left distributivity law (a, 13 E or3U) x*(a+/3)=y*n*(a+/3)=y*(rl*a+rl*f3+[a,/31y2'1) =x*a+x*13+ [a, P ](y217)y 288 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA with H(x) = (Y2 77)y. Here y2 is the James-Hopf invariant and [,] is the Whitehead product; compare Lemma 6.15.2. For the computation ofy2,q we consider the adjoint iq: M(77/2,3) -1153 =J(S2) of 17 where J(S2) is the infinite reduced product of James with 4-skeleton S2 UW e4. Here w = [i2, i2] is the Whitehead square. Hence i yields a map is S3 U2 e4 _ S2 U. e4 which extends the Hopf map 712: S3 - S2. Since 2712 = w we see that 1is a principal map between mapping cones associated with S3 I - S3

21 1w S3_--- S2 I2

This implies that q = y217: M(77/2,4) -> S5 is the pinch map. In factY2 -q is the composite of i and the James map J(S2) -J(S4), which is of degree 1 in dimension 4. Therefore H in (9.2.5) (1) carries x = 71y to qy E Ext(A,Z). Using the isomorphism (9.2.5) (3) the element x corresponds to41 E Hom(G(A), Z/4) withii = 7r4(Z/2, y) and µ(40 = H4(y) E Hom(A, Z/2). Let (y yo): dA -> d2 be a chain map representing H4(y) = µ(4i). Then qy E Ext(A,77) is represented by y, e Equivalently we have qy = aµ(41) and hence we get H = Bµ in A,(A). We clearly have P = 0 since S3 is an H-space.

We are now able to characterize the functor ir43> in (9.2.2) similarly as in Theorem 8.2.10.

(9.2.7) TheoremLet AG: G -> AAb be the functor which carries B E G to the inclusion AG(B): B ®1/2 ® A2(B) C G(H) ® A2(B) which is a A-quadratic map. Then there is an isomorphism [M(A,4), M(B, 3)] = 1r43)(A, B) = G(A, AG(B)) which is natural in A, B E G and which is compatible with A and µ in the universal coefficient sequence. Here d is the bifunctor in Definition 9.1.8. We point out that the isomorphism in Theorem 9.2.7 is available for all abelian groups A, B. Proof of Theorem 9.2.7If A or B are finitely generated we obtain the following commutative diagram which isnatural in M(A,4) E M4 and M(B,3) E M3. 16A B ®A0(A) B ®ir4 (S3) 16A° B ®Hom(A,Z/2)

lk 11 Ext(A,ir5(B,3))r--° - r4(A,M(B,3)) µ 10 Hom(A,B®l/2) 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA 289

Using Proposition 9.2.6 as an identification the top row is defined by applying the quadratic tensor product to the short exact sequence (9.1.5); compare also Definition 9.1.8 (1). We define k in the diagram by composition of maps, that is, for b, b' E B = n), a E ir4(A, S3), and c E ir4(A, S5) let

k(b ®a) = b o aandk([b, b'] ®c) = [b, b'] o c. On the right-hand side [b, b'] denotes the Whitehead product. The bottom row is the universal coefficient sequence where we have the identification ir4 M(B, 3) = B ® ZL/2 and

ir5M(B,3) = zr5(B,3) = G(B) ®A2(B); see Theorem 9.2.4. We obtain by A: B 0 Z/2 >-> G(B) and e in (9.1.7) the composite A * e,

B®AO(A) Ext(A,B®71/2(DA2B) Ext(A,7rs(B,3)) where A * = Ext(A, A ® A2B). Now one can check that the diagram above commutes. Since the rows are exact this diagram is actually a push-out diagram. This proves the theorem if A or B are finitely generated. In the general case we proceed similarly as in the proof of Theorem 8.2.10.

We have the commutative diagram

(9.2.8)

Ext(A,G(B)®A2B) 4* G(A,AG(B)) : Hom(A,B®71/2)

Ext(A,ir5M(B,3))>- > vr4(A,M(B,3)) -. Hom(A, ir4M(B,3))

The left-hand side is the isomorphism given by Theorem 9.2.4. The diagram is the metastable analogue of the corresponding stable result in (8.2.11). We now apply the push-out diagram (6.6.7) (ii) to the bottom row of (9.2.8). Thisyieldstheconnectionof the Moore bifunctor1i 3) withthe Eilenberg-Mac Lane functor H53). For the operator Q, n = 3, in Theorem 6.6.6 one has the simple description

ir5(B,3)=lr5M(B,3) H6K(B,3)=H6(B,3)

(9.2.9) II II G(B)®A2(B) µ -B*71/2®A2B where the bottom row is induced by the map µ in (9.1.2); see Theorem 9.3.5 290 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA below. Using (6.6.7) (ii) and (9.2.8) we now obtain the binatural push-out diagram with short exact rows

(9.2.10)

Ext(A, G(B) 0 A`B) H r43)(A, B) Hom(A, B ®7L/2) Ext(A,Q)J, push 1Q 11 Ext(A, H6(B,3)) N H53)(A,B)-µ,Hom(A, HS(B,3)) This push-out diagram in connection with Theorem 9.2.7 leads to the following computation of the Eilenberg-Mac Lane functor H53); compare (6.3.9), m = 3.

(9.2.11) TheoremThere is a binatural isomorphism (A, B E Ab) H531(A, B) = L#(A, B) 9 Ext(A, B *7L/2) where L,,,is the bifunctor in Definition 6.2.13. Moreover the isomorphism is compatible with 0 and µ.

Proof We combine the push-out (9.2.10) and Theorem 9.2.7. For this we need the composite Ext(A,µ91)O*B=Ext(A,(gA)ED 1)982 (1) where gA = 0. Hence HS (A, B) is the direct sum of Ext(A, B * Z/2) and the push-out of the top row in the following diagram Ext(A,A2B)B®A0(A) B®A,(A)

II 1Pr, 1Be (µ, 1) (2) Ext(A, A2B) (A2B) ®Ext(A,Z) B ®L(A) Here (A, 1): A1(A) - L(A) is the natural map between quadratic Z-modules given by A1(A) = (Hom(G(A),71/4)dµExt(A,Z) -°a Hom(G(A),7L/4))

IL 11 lµ L(A) =(Hom(A,7L/2)-'-* Ext(A,Z)-- Hom(A,7L/2)) compare the definition of A1(A) in (9.1.4). We now observe that the push-out of the top row of (2) is via diagram (2) isomorphic to the push out of the bottom row of (2). This follows since

Ext(A,7L/2) A1(A) - L(A) (3) 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA 291 is a short exact sequence of quadratic 7L-modules and since the quadratic tensor product is left exact. Now the push-out of the bottom row of (2) is L,,(A, B). Here we assume that A or B are finitely generated.

(9.1.2) Remark Using the quadratic A-map ,1=0®A2B: B®7L/2®A2B->B*7L/2®AFB we get the binatural isomorphism L,(A, B) = G(A, .1). Here the right-hand side is defined in Definition 9.1.8.

9.3 Whitehead's group F5 of a 2-connected space

We here compute the groups r5 X and r4(A, X) of a 2-connected space X and we determine the functorial properties of these groups; we proceed similarly as in the stable case; see Section 8.3. Since the groups depend only on the 4-type of X we have for 1, = r)3 : 1r3(X) ® 71/2 -p ir4 X

(9.3.1) r5(X) = r5K(1,,3) and

(9.3.2) F4(A, X) = r5(A, K(11,3)). The computation of these groups is based on the next result.

(933) PropositionLet X be a 2-connected space. Then one has the natural exact sequence

1r4(X) ®7 L/2 ® A2(H3X) N r5X -. H3(X)*7L/2.

Here A is given by the Hopf map 1]3 and by the Whitehead product that is O(a(& 1)=a113 for aca lr4(X)and O(xAy)=[x,y] for x,yE1r3X=H3 X. Proof The proof is similar to the proof of Proposition 8.3.5. We again consider the mapping cone Cg with n = 3. Here, however, we use the exact EHP sequence for the mapping cone C5; see Theorem A.6.9.

(93.4) CorollaryLet X be a 2-connected space. Then one has a short exact sequence

Az(H3X) N I5(X)11'6('X) 292 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA which is naturally split. Here I is the suspension operator and A is defined as in Proposition 9.3.3.

Proof The suspension maps the sequence in Lemma 9.2.3 surjectively to the sequence in Proposition 8.3.5. Moreover E: ir4X -> ir5Y' X is an isomorphism. Hence we obtain the required exact sequence. Now let 13: M(B, 3) -> X be a map which induces the identity H3 /3 with B = H3 X. Then Lemma 9.2.3 and Proposition 9.3.3 yield the push-out diagram

ir4X®71/2®A2B A I'SX

17,* 1 0.1 B®7L/2®A2B rr6M(B,4)ED A2B ar5M(B,3)

This shows that AZB is a direct summand of I'5X. Moreover Q in Theorems 6.6.6 and 6.6.11 yields a natural retraction of A in the Corollary. More generally than in Theorem 9.2.4 we now get the following result which is an unstable version of Theorem 8.3.7.

(9.3.5) TheoremLet X be a 2-connected space. Then there is a natural commutative diagram

1 Ext(Z/2, zr5X) ® A2173X 0ir4(7L/2,X) ®A27r3Xµ®0Hom(7L/2, ir3X) Iltt w> II II µ i5(X) ® Z/2 ® A2H3X -) F (X) H3(X) *7L/2

Here the top row is given by the universal coefficient sequence for 1T4(7L/2, X). The left- and the right-hand side denote the canonical identifications. Moreover, we obtain for 17 in (9.3.1) the isomorphism of groups

0: G(-q) ® A2(H3X) = F(X) which is compatible with A and A. Here G(q) is the group in Definition 8.1.3 (A).

Proof The isomorphism Q*, w) is defined by the map C* in Theorem 8.3.7 and by the Whitehead product w in Proposition 9.3.3 above.

(93.6) Remark By setting X = K(B, 3) we readily get

H6K(B, 3) = I'5K(B,3) = B *7L/2 ® AZB.

Moreover since /3 is used for the definition of the operator Q in Theorem 9 2-CONNECTED 6-DIMENSIONAL POLYHEDRA 293

6.6.6, we deduce from the diagram in the proof of Corollary 9.3.4 that diagram (9.2.9) commutes.

We now study the group F4(A, X) in a similar way as in Lemma 8.3.9.

(93.7) LemmaLet X be 2-connected with B = H3 X and let /3: M(B, 3) --> X be a map which induces the identity H3( /3 ). Then one has the commutatuwe diagram

Ext(A, FSX)>') F4(A, X) Hom(A,F4X)

TExt(A.0,) ja. I (r4JS). Ext(A, 7r5M(B,3)) - a4(A, M(B,3)) -µ» Hom(A, B (9 71/2)

The top row is the universal coefficient sequence (Definition 2.2.3) which is natural in X. For X = M(B, 3) the top row yields the bottom row which also coincides with the exact sequence in (9.2.8). Since the diagram is a push-out diagram we can compute the group I'4(A, X) by use of (9.2.8) and Theorem 9.3.5. A homomorphism rl: B (& Z/2 -> it yields the quadratic A-map

(9.3.8) A=O(rl® 1)®AZB: B®Z/2®MB - G(rl)®AFB =L given by O(i(9 1):

(9.3.9) TheoremFor the abelian group A and for lj: B 0 Z/2 - it one has the isomorphism

0': G(A, O(r, (9 1) (D AZB) = r4(A, K(-q, 3)) which is compatible with 0 and µ.

Compare the isomorphism in (8.3.10). The theorem extends Theorem 9.2.7. 10 DECOMPOSITION OF HOMOTOPY TYPES

In this chapter we describe explicit results on the classification of homotopy types. For example we give a complete list of (n - 1)-connected (n + 3)- dimensional homotopy types X, n >- 4, for which all homology groups H;(X) are cyclic. We also obtain a complete list of all homotopy types X for which 1rn X, 7r + i X, 7r + 2 X, n >- 4, are cyclic groups and for which 7ri X is trivial otherwise. These results are obtained by describing the corresponding indecomposable homotopy types. The indecomposable (n - 1)-connected (n + 3)- dimensional polyhedra were found in Baues and Hennes [HC]. The complete solutions of classification problems in this chapter show that the prospect of homotopy types is not so dim.

10.1 The decomposition problem in representation theory and topology

Let C be a category with an initial object * and assume sums, denoted by A V B, exist in C. An object X in C is decomposable if there exists an isomorphism X = A V B in C where A and B are not isomorphic to * . Hence an object X is indecomposable if X -=A V B implies A or B = * . A decomposition of X is an isomorphism (10.1.1) in C where A; is indecomposable for all i E (1,..., n). The decomposition of X is unique if B, V ... v Bm - X = A, V V A,, implies that m = n and that there is a permutation o, with B,.A. A morphism f in C is indecomposable if the object f is indecomposable in the category PaIr(C). The objects of Pair(C) are the morphisms of C and the morphisms f -p g in Pair(C) are the pairs (a, f3) of morphisms in C with ga = /3f. The sum of f and g is the morphism f V g = (i, f, i 2 g). The decomposition problem in C can be described by the following task: find a complete list of indecomposable isomorphism types in C and describe the possible decompositions of objects in C. We now consider various examples and solutions of such decomposition problems. These examples originated in representation theory and topology. First let R be a ring and let C be a full category of R-modules (satisfying some finiteness restraint). The initial object in C is the trivial module 0 and the sum in C is the direct sum of modules, denoted by Me N. With respect to the decomposition problem for modules in C, Gabriel states in the 10 DECOMPOSITION OF HOMOTOPY TYPES 295

introduction of [IR]: `The main and perhaps hopeless purpose in representation theory is to find an efficient general method for constructing the indecomposable objects by means of simple objects, which are supposed to be given'. Various results on such decomposition problems are outlined in Gabriel [IR]. We shall use only the following examples.

(10.1.2) ExampleFor R = 77 let C be the category of finitely generated abelian groups. In this case the indecomposable objects are well known; they are given by the cyclic groups 77 and Z/p' where p is a prime and i >_ 1.

(10.1.3) ExampleLet k be a field and let R be the quotient ring R = k (X, Y)/(X 2, y2). Here (X 2, Y2) stands for the ideal generated by X2 and y2 in the free associative algebra k (X, Y)in the variables X and Y. Let C be the full category of R-modules which are finite dimensional as k-vector spaces. C.M. Ringel [RT] gave a complete list of indecomposable objects in C. These objects are characterized by certain words which are partially of a similar nature as the words used in Section 10.2 below.

(10.1.4) ExampleIn topology we also consider graded rings like the Steen- rod algebra and graded modules like the homology or cohomology of a space. Let R = 21 P be the mod p Steenrod algebra and let k >_ 0. We consider the category C of all graded R-modules H for which H; is a finite Z/p-vector space and for which H; = 0 for i < 0 and i > k. It is a hard problem to compute the indecomposable objects of C; only for k< 4p - 5 is the answer known by the work of Henn [CP]. In fact, Henn's result is closely related to the result of Ringel in Example 10.1.3 above; to see this we consider the case p = 2. The restriction k < 3 then implies that the 21-module structure of H is completely determined by Sq, and Sq2 with Sq, Sq, = 0 and Sq2 Sq2 = 0. Hence, forgetting degrees, the module H is actually a module over the ring 77/2(X, Y)/(X 2, Y 2) with X = Sq1, Y = Sq2 and such modules were classified by Ringel.

Next we describe the fundamental decomposition problem of homotopy theory. Let Top*/= be the homotopy category of pointed topological spaces. The set of morphisms X - Y in Top*/= is the set of homotopy classes [X,Y]. Isomorphisms in Top*/= are called homotopy equivalences and isomorphism types in Top*/= are homotopy types. Let An be the full subcategory of Top*/=consisting of finite (n - 1)-connected (n +k)- dimensional CW-complexes; the objects of A' are also called At-polyhedra, see J.H.C. Whitehead [HT]. The suspension Y_ gives us the sequence of functors

F (10.1.5) A; AZ ... -A Ak+1 - ... which is the k-stem of homotopy categories. The Freudenthal suspension 296 10 DECOMPOSITION OF HOMOTOPY TYPES theorem shows that for k + 1 < n the functor 1: A,, -> Ak+ Iis an equivalence of categories; moreover for k + 1 = n this functor is full and a 1-1 correspondence of homotopy types. We say that the homotopy types of AR are stable if k + 1 < n; the morphisms of An, however, are stable if k + 1 < n. The computation of the k-stem is a classical and principal problem of homotopy theory which, in particular, was studied for k < 2 by J.H.C. Whitehead [SC], [HT], [CE]. The k-stem of homotopy groups of spheres, denoted by 7n+k(S"), n z 2, is now known for fairly large k; for example one can find a complete list for V<, 19 in Toda's book [CM]. The k-stem of homotopy types, however, is still mysterious even for very small k. The initial object of the category AR is the point * and the sum in A,, is the one-point union of spaces. The suspension I in (10.1.5) carries a sum to a sum and E: Ak -> yields a 1-1 correspondence of indecomposable homotopy types for k + 1 < n. As in the case of modules we use a finiteness restraint: we consider the decomposition problem in the stable k-stem of homotopy categories only for finite (or equivalently compact) CW-complexes. The following results on the decomposition problem in the category AR are known. Recall that the elementary Moore spaces of AR are the spheres S"', n < m < n + k, and the Moore spaces M(71/p', m) where p' is a prime power and n < m < n + k. These are indecomposable objects in A. The next result is classical and follows by use of the Hurewicz theorem from Example (10.1.2).

(10.1.6) Proposition (A) For n >_ 1 the sphere S" is the only indecomposable homotopy type of A° and each object in A°, has a unique decomposition.

(B) Let n >_ 2. The elementary Moore spaces of A' are the only indecomposable homotopy types in A;, and each object in At,, has a unique decomposition.

It is known that there are 2-dimensional complexes in A, which admit different decompositions, see for example Dyer and Sieradski [TH]. Next we consider the decomposition problem in the category A2, n >_ 3. For this we define in the following list the elementary complexes X of Chang which are mapping cones of the corresponding attaching maps in the list. Let i1, resp. i2 be the inclusions of Sn+', resp. S", into the one-point union S"+' V S" and let ibe the Hopf map and p, q be powers of 2.

(10.1.7) Elementary complexes of Chang X attaching map Sn+I - Sn X(71)=S"Ue"+2 fin' X67q)=S"vS"+'Uen+2 qiI +i2T)n: Sn+I_ Sn+IVSn Sn+' VSn-) Sn X(F17) = S" U e"+ 1 Uen+2 (i,p): X(71q)=S" VSn+l Uen+' Uen+2(qi1 +i2r1,,,Pi2): Sn+I V V Sn 10 DECOMPOSITION OF HOMOTOPY TYPES 297

These complexes are also discussed in the books of Hilton [IH], [HT]. Our notation of the elementary Chang complexes above in terms of the `words' 7r, 71q,,71,,?7q is compatible with the notation on elementary A3,-complexes in Section 10.2. These words can also be realized by the following graphs where vertical edges are associated with numbers p, q and where the edge connecting level 0 to 2 is denoted by 71.

bUd lid bit It

Equivalently these are all subgraphs (or subwords) of p,lq which contain 7r. In Section 10.2 we shall describe the elementary by subgraphs (or subwords) of more complicated graphs.

(10.1.8) Theorem of Chang [HI]The elementary Moore spaces and the elementary Chang complexes above are the only indecomposable homotopy types in A;,, n >_ 3, and each homotopy type in A;, has a unique decomposition.

Proof We use Theorem 3.5.6 which shows that each homotopy type X in A2 n >_ 3, is given by a (&Z/2-sequence H, b H (9 71/2 ` 7r -h -, Ho - 0 (1) with H = H X, Ho = Hn +, X, H, = Hn+ 2X finitely generated, H,free abelian, and 7r= 7rn+,X. The elementary Moore spaces and the elementary Chang complexes yield the G Z/2-sequences in the following list where p and q are powers of 2 and I is a power of an odd prime. -b - X H, H®1/2`-I, IF _L Ho H I Sn -4 7/2 Z/2 - 2 Sn+I -a 0 ------41 z 0 sn+ 2 - 0 -a 0 -a 0 M(7L/q, n) Z/2 2/2 71 /q M(7L/q, n + 1) 0 71/q 0 M(7L/1, n) 0 0 Z/1 M(71/l, n + 1) --a 0 7L/l 0 X(,7 ) Z/2 0 z 2 X(719) 7L/2 7L/2q 7 X(pr1) 7/2 0 71/P 2 X(p7lq) Z/2 7L/2q 71/P 298 10 DECOMPOSITION OF HOMOTOPY TYPES

We have to show that each sequence (1) is a direct sum of sequences as in the list. Then the additive detecting functor A' in Theorem 3.5.6 yields the proposition in Theorem 10.1.8. We choose a basis B(ar) c ar where B(ar) yields a decomposition of ar as in Example 10.1.2. For x E B(ar) we get hx E Ho and the orders satisfy eitherI hxl = Ixl S - orI xI = 2Jhxl = 2q+'. Moreover the elements x in the second case yield a system of generators 2qx of image i. The elements hx with x E B(a) and hx * 0 form a basis of Ho. We choose a basis B(H) c H1 and B(H ® 71/2) c H ® 71/2 such thati carries elements of B(H (9 1/2) to elements of the form 2qx, x e B(ar), and such that b carries an element y E B(H) to an element in B(H (9 Z/2) or to 0. Using these generators all homomorphisms b, i, h are given by diagonal matrices. Finally we choose a basis B(H ® Z/2). These generators now yield a direct sum decomposition of (1) such that each summand is one of the sequences in the list above. Moreover this decomposition is unique in the sense of (10.1.1). Spanier-Whitehead duality yields the following equations which we easily derive from the definitions: (10.1.9) PropositionThe Spanier-Whitehead duality functor D: AR - A2 satisfies DX(q) =X(aq) DX(,qq) =X(9,1) DX(prl) =X(ip) DX(palq) =X(gfP). Hence the Spanier-Whitehead duality turns the graphs in (10.1.7) around by 180°; see also Definition 10.2.3 and Theorem 10.2.10 below.

10.2 The indecomposable (n - 1)-connected (n + 3)-dimensional polyhedra, n >- 4

We here describe all elementary (n - 1)-connected (n + 3)-dimensional polyhedra in terms of certain words, or graphs. This extends the results of Chang in Theorem 10.1.8. The fundamental results in this section are deduced from the classification of stable (n - 1)-connected (n + 3)-dimensional polyhedra in Chapter 8; compare Baues and Hennes [HC]. For the description of the indecomposable objects in A3.,n z 4, we use certain words. Let L be a set, the elements of which are called `letters'. A word with letters in L is an element in the free monoid generated by L. Such a word a is written a= a1a2 a with a; E L, n z 0; for n = 0 this is the empty word 46. Let b = b 1 bk be a word. We write w = ... b if there is a 10 DECOMPOSITION OF HOMOTOPY TYPES 299 word a with w = ab, similarly we write w = b if there is a word c with w = be and we write w = ... b if there exist words a and c with w = abc. A subword of an infinite sequence a a _ , a0 a, a2 with a; E L, i E Z, is a finite connected subsequence anan an+k, n c- Z. For the word a = a, ... a we define the word -a = an an a, by reversing the order in a.

(10.2.1) Definition We define a collection of finite words w = w,w2 W. The letters w; of w are the symbols f, 77, e or natural numbers t, s;, r;, i E Z, which are powers of 2. We write the letters s; as upper indices, the letters r, as lower indices, and the letter t in the middle of the line since we have to distinguish between these numbers. For example 774f 2718 is such a word with t = 4, r, = 8, s, = 2. A basic sequence is defined by

S11r, cS27lr, (1) This is the infinite product a(1)a(2) of words a(i) = i > 1. A basic word is any subword of (1). A central sequence is defined by

175 f'27"2 (2) A central word w is any subword of (2) containing the number t. Whence a central word w is of the form w = atb where -a and b are basic words. An e-sequence is defined by

5 6r-"qs (3) An e-word w is any subword of (3) containing the letter e; again we can write w = aeb where -a and b are basic words. A general word is a basic word, a central word, or an a-word. A general word w is called special if w contains at least one of the letters 77, or a and if the following conditions (i), DO), (ii), and D(ii) are satisfied in case w = aeb is an a-word. We associate with b the tuple

( (s...... sm, ifb = t= s(b) = (sb, sb,...) = St(s,...., otherwise

ifb = 17 r,, otherwise where s, sn, and r, rr are the words of upper indices and lower indices respectively given by b. In the same way we get s(-a) = (s,-°, s2 , ...) and r(-a) = (r, °, r2 °,..) with S ° E (s;, x,1) and r,- ° E {r_1, x, 1), i e N. The conditions in question on the a-word w = aeb are: (i) b=4-a062

D(i) a=(k -b#277. 300 10 DECOMPOSITION OF HOMOTOPY TYPES

Moreover if a * 4) and b * 4) we have

(ii) s,=2=* r_1>_4

and

(2r b,, -S2r2,-S3,r3,...,-Si,r,,b b b b b b .. < (rl -S,r2-s2a, r3 °,-S3 °,..r, °, -S! °,... )

D(ii) r_ 1 = 2 s, >_ 4 and

(-sb, rb, -sz, rz, -s ,rj, ... I-sb, rb,.. ) <2'S,°,r2°,-S2°,r3°,-S3°,...,ri °,-S1 °, ..).

The index i runs through i = 2,3.... as indicated. In (ii) and D(ii) we use the lexicographical ordering < from the left, that is (n,, n2, ...) < (m,, m2,...) if and only if there is t >: 1 with nj = mi for j < t and n, < m,. Finally we define a cyclic word by a pair (w, (p) where w is a basic word of the form (p >- 1)

W = S eS2,gr... e SP (4) 71, 77r, and where cp is an automorphism of a finite dimensional 1/2-vector space V = Two cyclic words (w, cp) and (w', cp') are equivalent if w' is a cyclic permutation of w, that is,

w' = tsi'Ir, ... 01Ir, fs1 Ir'... 6Sj'7r(-', and if there is an isomorphism 'I': V(rp) = V(cp') with p = Y'-' V. A cyclic word (w, cp) is a special cyclic word if p is an indecomposable automorphism and if w is not of the form w = w'w'w' where the right-hand side is a j-fold power of a word w' with j > 1. The sequences (1),(2),(3) can be visualized by the infinite graphs sketched below. The letters s;, resp. r;, correspond to vertical edges connecting the levels 2 and 3, resp. the levels 0, 1. The letters 71, resp. 6, correspond to diagonal edges connecting the levels 0 and 2, resp. the levels 1 and 3. Moreover e connects the levels 0 and 3 and t the levels 1 and 2. We identify a general word with the connected finite subgraph of the infinite graphs below. Therefore the vertices of level i of a general word are defined by the 10 DECOMPOSITION OF HOMOTOPY TYPES 301 vertices of level i of the corresponding graph, i E (0, 1, 2, 3}. We also write JxJ = i if x is a vertex of level i.

3 2

basic sequence

3 S-1 SI 2 IV r_3 -' rl rl 0 central sequence

3 s_Z s_1 S1 2

r 2 r-t rl 0 E-sequence

(10.2.2) Remark There is a simple rule which creates exactly all graphs corresponding to general words. Draw in the plane l2a connected finite graph of total height at most 3 that alternatingly consists of vertical edges of height one and diagonal edges of height 2 or 3. Moreover endow each vertical edge with a power of 2. An equivalence relation on such graphs is generated by reflection at a vertical line. One readily checks that the equivalence classes of such graphs are in 1-1 correspondence to all general words.

(10.2.3) DefinitionLet w be a basic word, a central word, or an a-word. We obtain the dual word D(w) by reflection of the graph w at a horizontal line and by using the equivalence defined in Remark 10.2.2. Then D(w) is again a basic word, a central word, or an e-word respectively. Clearly the reflection replaces each letter f in w by the letter 71 and vice versa; moreover it turns a lower index into an upper index and vice versa. We define the dual cyclic word D(w, q) as follows. For the cyclic word (w, (p) in (10.2.1) (4) let D(w, cp) = Here we set

W' = 6"1752 f r2 ... sv t rp%, and we set q'= Hom(c,7L/2) with V(cp) Up to a cyclic permutation w' is just D(w) defined above. We point out that the dual words D(w) and D(w, (p) are special if and only if w and (w, cp) respectively are special. 302 10 DECOMPOSITION OF HOMOTOPY TYPES

As an example we have the special words w =2r14e2ij8Sto4,q and D(w)_ 54_q862-1452 whichare dual to each other. They correspond the graphs

1 0

W =

I 0 0(W) =b417%,742 Hence the dual graph D(w) is obtained by rotating the graph of w. We are going to construct certain A3,-polyhedra, n >_ 4, associated with the words in Definition 10.2.1. To this end we first define the homology of a word.

(10.2.4) DefinitionLet w be a general word and let rQ rr and sµ ... s be the words of lower indices and of upper indices respectively given by w. We define the torsion groups of w by To(w)=7L/r, 6) 71/rs, (1)

T,(w) = 7L/tif w is a central word, (2) T2(w)=7L/sµ® ®7L/s,,, (3) and we set Ti(w) = 0 otherwise. We define the integral homology of w by Hi(w) = Z',(*) ED Ti(w) 6) ZR,(w). (4) Here /3;(w) = L;(w) + R;(w) is the Betti number of w; this is the number of end-points of the graph w which are vertices of level i and which are not vertices of vertical edges; we call such vertices x spherical vertices of w. Let L(w), resp. R(w), be the left, resp. right, spherical vertex of w in case they occur. Now we set L;(w) = 1if IL(w)I = i and R;(w) = 1if IR(w)I = i; moreover L;(w) = 0 and R;(w) = 0 otherwise. Using the equation (4) we have specified an ordered basis B; of H;(w). We point out that Po(w) + /3,(w) + 132(w) + 93(W) S 2. (5) For a cyclic word (w, gyp) we set

H,(w, gyp) _ ® T(w) (6) 10 DECOMPOSITION OF HOMOTOPY TYPES 303 where v = dim V('p) and where the right-hand side is the v-fold direct sum of T(w). As an example we consider the special words

w =02 1184 w _2n844iJ16 The homology of these words is:

W=8 327)8 w'=2 778 5 41716 H3 z 0 H2 1/32 Z/4 H1 0 71/2 Ho 71®1/8 Z/8,@ 7l/16 Here w has two spherical vertices while w' has no spherical vertex. We point out that the numbers 2k attached to vertical edges correspond to cyclic groups Z/2kin the homology. We describe many further examples in Sections 10.5 and 10.6. For the construction of polyhedra X(w) associated with words w we use the following generators; see Definition 11.6.3 below.

(10.2.5) Generators of homotopy groupsLet r, s be powers of 2. We have the Hopf maps

77 _ . Sn+1 Sn . Sn+2 Sn+1 e=2: Sn+2--aSn.

We use the composites Sn+l -M(71/r,n), 71=177n: 6=77n+19: M(71/r,n + 1) --Sn+l which are (2n + 1)-dual. Moreover we have the (2n + 2)-dual groups, n >_4

for r=2 [Sn+z, M(71/r, n)]= {l/2+i/2e, forr4

71/417z for s = 2 [M(7L/s,n+1),SR]= Z/2775+7L/2esfors-4

where e, = i-q and es = il,and, =XZJzand 775 = 772Xz Next we use

®71/2r1z fors=r=2 Z/441' L/s,),(M(lln+1M /r,n)J-1 Q 2 71/2 g, ®71/ 4Fir fors=2I r>4 7L/2 f; ® 71/271,5 9 ll/2 e, otherwise. 304 10 DECOMPOSITION OF HOMOTOPY TYPES

Hence we have Irs = X2 f2 q, rl; = i'n2X2, and e, = We have the (2n + 2)-dualities D(,') _ -q5 and D(e,) = es.

(10.2.6) DefinitionLet n >_ 4 and let w be a general word. We define the A3,-polyhedron X(w) = Cf by the mapping cone Cf of a map f = f(w): A - B where A = M(H3, n + 2) v M(H2, n + 1) V Sn+' B=M(Ho,n)VS.+'VSb+' (1) Here Hi = Hi(w) is the homology group in Definition 10.2.4 above. We set S +' = S"+' if w is a central word and we set SC+' otherwise; moreover we set Sb+'= S"+' if w is a basic word of the form w =... and we set Sb ' otherwise. We now obtain the attaching map f=f(w): M(H3,n+2)VM(H2,n+1)vSC+'-*M(Ho,n)V SC+' VSb+'

(2) as follows. We first describe B and A in (1) as one-point unions of elementary Moore spaces. For each letter rs of r, rr (see Definition 10.2.4) we have the inclusion j(rs): M(77/rs,n)cB. (3) Moreover for each spherical vertex x of w with 41 :5 1 we have the inclusion j(x): S"+lxl cB. (4) This is the inclusion of Sb+' if Ixl = 1. The space B is exactly the one-point union of the subspaces (3), (4) and of j,: S°+' cB. Next we consider the space A in (1). For each letter s, of sµ s (see Definition 10.2.4) we have the inclusion j(s,): 1) cA. (5) Moreover for each spherical vertex x of w with Ix1 >_ 2 we have the inclusion j(x): S""xH-' cA. (6) The space A is exactly the one-point union of the subspaces (5), (6) and of jc: Sn'' cA. We now define f = f(w) by the following equations. For a letter s, as above and for 6 = T - 1 we set

j(rs)rfi +1(r,)ij' if W= .rn s'71r,... j(rs)rla +j(rr)r;' if w= ...ryr)5'fr,... fj(s,) _ (7) jcfl"+,q +j(r, )7l ;' if w= t e 3 7 r a n d j(r_,)e;' +j(rl)71;'if w = r_ Ies'l7r and T= 10 DECOMPOSITION OF HOMOTOPY TYPES 305

The first equation also holds if the letters rs are empty, that is if w = f s-77 or if w = 65'71 respectively. In this case we set j(rs) =j(x), if x = L(w), resp. j(rT) =fly), if y = R(w); see Definition 10.2.4. We use a similar convention for the other equations in (7). Using (2) and (7) we see that fj(sT) is well defined for all general words w. Next we define fj(x) where x is a spherical vertex of w with Ixl =2.

j(ra)4Q ifw = r;,o..., Ixl = 3, x =L(w) j(ra)i71n if w =71r....,Ixl = 2,x = L(w) fj(x) = if w= (8) rE,Ixl=3,x=R(w). Using (8) and (2) the element fj(x) is well defined for all general words w. Finally we define fjr by

fj, j(r-I )i7jn +jr(h)if w = ... r71t ... = j(x)'gn +j,(ti) if w = 71t..., x =L(w) (9) j,(ta) if w=t" .

Here a is the identity of S". This completes the definition of f = f(w) and hence the definition of X(w) = Cf.

We point out that the construction of f(w) follows exactly the pattern given by the word w or the graph of w. For this we subdivide the graph of w by a horizontal line between levels 1 and 2; all edges crossing this line are summands in the attaching map f(w). For example consider the graphs E32718 C, 2% i; 47116, and 27146 2,n864,n above. Then we get

M(7L/32, n+ 11)) v Sn e 2 t IE f(e3271a6)= I e S" v M(7L/8, n)

Sn+ I V M(7L/4, n + 1)

rr _ f(27I86 47716)- 12 16 S"+ I v M(l/8, n) V M(7L/16, n)

S"+ t v M(7L/2, n+ 1) V M(7L/4, n+ 1)

tt tt f(2 745 2718 5 471)= li'7 If if M(l/2, n) V S"+ I V M(71/8, n)v S"

Here , 71, e are the corresponding generators in (10.2.5). 306 10 DECOMPOSITION OF HOMOTOPY TYPES

(10.2.7) DefinitionLet n > 4 and let (w, (p) be a cyclic word. We define the A3,-polyhedron X(w, cp) = Cf by the mapping cone of a map f = f(w, (p) where f: M(H2,n + 1) -*M(Ho,n) (1) with H; = Hi(w, cp); see Definition 10.2.4 (6). For u e (1....,v} we have the inclusion (m = n, n + 1 and i = 0, 2) j,,: M(T(w),m) cM(H,,m) (2) by the direct sum decomposition in Definition 10.2.4 (6). Moreover we have for each letter rs and s., of r, r,, and s, sp (see Definition 10.2.1 (4)) the inclusions j(rs): M(7L/2rs,n) cM(T0(w),n), (3) j(s,,): M(7L/25-, n + 1) c M(T2(w), n + 1). (4) Compare (10.2.5) (3) and (5). We choose a basis (b1,...,b1.} of the vector space V(qp) and we define < E {0, 1) by E" .I (p,,be. This yields a definition of f by the following formulas (5) and (6).

+j(rr)rl, ] (5)

If w = ,3 6 T E (2,..., p), and 6 = r - 1; see Definition 10.2.1 (4). Moreover we set

(Pujej(rp)Sro'. (6) e=1 The spaces X(w) and X(w, gyp) are constructed in such a way that the integral homology is given by

(10.2.8) H,,,X(w) = Hi (w), (p) = H1(w, (p) where we use the homology of the words w and (w, (p) in Definition 10.2.4.

The next result solves the decomposition problem in the category A3 n >_ 4. This result generalizes the theorem of Chang (10.1.8) for the next dimension. Its proof, however, is considerably more intricate than the fairly direct proof of Theorem 10.1.8 above.

(10.2.9) Decomposition theoremLet n >_ 4. The elementary Moore spaces in A',, the complexes X(w) where w is a special word, and the complexes X(w, q,) where (w, qp) is a special cyclic word furnish a complete list of all indecomposable homotopy types in A',. For two complexes X, X' in this list there is a homotopy equivalence X = X' if and only if there are equivalent special cyclic words 10 DECOMPOSITION OF HOMOTOPY TYPES 307

(w, q,) - (w', gyp') with X = X(w, (p) and X' = X(w', gyp'). Moreover each homotopy type in Art has a unique decomposition.

The proof of the decomposition theorem relies on the classification theorem (8.1.6). Given this theorem one can solve the decomposition problem in the algebraic category of A3-systems. For this an intricate generalization of the representation theory of Ringel and Henn is needed; see (10.1.3) and (10.1.4). We refer the reader to Baues and Hennes [HC] for the complete proof of the decomposition theorem. Spanier-Whitehead duality of indecomposable complexes in A3 is completely clarified by the next result.

(10.2.10) TheoremLet n >_ 5. For a general word w and for a cyclic word (w, q') let Dw and D(w, (p) be the dual words defined in Definition 10.2.3. Then X(Dw) is the Spanier-Whitehead (2n + 3)-dual of X(w) and X(D(w, (P)) is the Spanier-Whitehead (2n + 3)-dual of X(w, cp).

Proof The result essentially follows from the careful choice of generators in (10.2.5) which is compatible with Spanier-Whitehead duality. This implies that there are (2n + 2)-dualities f(w)* =f(Dw) and f(w, (p)* =f(D(w, gyp)). Hence the proposition is a consequence of the fact that Spanier-Whitehead duality carries a mapping cone of f to the mapping cone of D(f ). We point out that X(w) in (10.2.5) coincides with the corresponding elementary complex in (10.1.7) if w is one of the words n, ijq, P71, p77q. Moreover the suspensions of such complexes are given by

(10.2.11) Y.X(rl) =X(C), IX(qq) .X(P71) =X(pi; ), EX(p7lq) The words prlq and pf4 correspond to the two possible subgraphs in a central sequence which both look like the graph in (10.1.7). Hence the Chang complexes yield only the following elementary A3,-polyhedra

3- q 2- 1- 0- PT19 P44 This precisely describes the embedding of indecomposable Am-polyhedra (m = n, n + 1) into the much larger set of indecomposable A,3,-polyhedra. In a similar way we expect horrendous complexity if one considers the embedding of indecomposable (m = n, n + 1) into the unknown set of indecomposable AR-polyhedra. As a corollary of the decomposition theorem we get the following surprising result. 308 10 DECOMPOSITION OF HOMOTOPY TYPES

(10.2.12) TheoremLet n >- 4 and let X be an (n - 1)-connected (n + 3)- dimensional finite polyhedron with Betti numbers /3,(X). If 2<0n(X)+Yn+1(X)+10n+2(X)+Rn+3(X) or if Hn + 1(X) contains the direct sum of two cyclic groups then X is decomposable. Proof A general word has at most two spherical vertices and hence an indecomposable homotopy type X in A;, satisfies dim(H,, X ® Q) < 2. More- over Hn+ 1wis non-trivial only for central words w and general words w = 6 ... and in these cases Hn+ 1w is Z/2' or 71 respectively. Hence, X is indecomposable, implies H,,+ 1 X is cyclic of prime power order or Z. (10.2.13) The A'-system of X(w) By Theorem 8.1.6 we have the detecting functor A': spaces -A 3-System. This is an additive functor between additive categories. Hence the indecomposable complexes X(w) in the decomposition theorem 10.2.9 correspond via A' to indecomposable A3-systems S(w) = A'X(w). (1) We here compute the A3-system S(w) = (Ho,H2,H3,1 1,bz,rlw,b3,/w) (2) explicitly in terms of w. Here we have the homology H, = H,(w) = Hn+r(X (w)),i E (0,1, 2, 3), (3) as defined in Definition 10.2.4 and we have the homotopy groups Tr, _ ir,(w) _ ?rn+,(X(w)),i E {1,2}, (4) which are part of the exact F-sequence of S(w) given by w as in Definition 8.1.5:

H3 G(rt) --> 2 -- H, H®® 71/2 -'-> ir, -L H1 -* 0. (5) This sequence is isomorphic to the corresponding part of Whitehead's exact sequence for X(w); see (8.1.7). We first describe bZ; for this we denote the basis of Hi(w) in Definition 10.2.4 (4) by B0 = (L0(w),r,,,...,rr, Ro(w))cHo B1 = (L,(w),t,R1(w)) cH, (6) B2 = {L2(w), sµ..... s.., R2(w)} cH2 B3 = (L3(w), R3(W)) cH3. 10 DECOMPOSITION OF HOMOTOPY TYPES 309 We also write Lo(w) = rQ_ 1, Ro(w) = r,,+, and we set L2(w) = sµ_, and

R2(w) = Iin case there are spherical vertices of w; see Definition 10.2.4. With this notation we define bz:H,- H0®71/2 (7) rs®1 a b;' (s,r) =rr ® 1 if w = ... 0 otherwise. Hence bz carries basis elements to basis elements or to the trivial element so that the cokernel of bz is cok(bz) = ED {(Z/2)rr,rr(9 1 e image bz). (8) We now define 7r1(w) by

7T' (71/20i ® cok(bw)/(Z/2)r_, if w= r77t (9) H, ® cok(b; otherwise with H, = H1. The homomorphism -q' in (5) is given by the composite

,q': Ho ®71/2_*cok(b,)clr, (10) where the inclusion i carries the basis element to tiif w= r _ lrit and carries it to Pr otherwise. Clearly h in (5) is trivial on the second summand of 7r,(w) and satisfies h(i) = t and h I H, = id respectively. Now (10) induces the homomorphism r7' 0 1: Ho ® 71/2 - ar, ® 71/2 (11) which carries r_ 1 ® 1 e image b"' to the trivial element if w = r_ , r7t and carriesrr ® 1 t image b2'to1'r ® 1 otherwise. Hence qw (9 1carries basis elements to basis elements or to the trivial element. Using (11) we obtain the group G67') together with the short exact sequence

7r,0 Z/2° G(rl"')-'Ho *7/2 (12) by the push-out of Definition 8.1.3. The basis of Ho * Z/2 is {rr/2, a < r< B) and we have the basis elements r,/2EG67w)with µ(rr/2/2)=rr/2. (13) Here the order of rr/2 is 4 if the number rr is 2 and if (ri' 0 1Xrr 0 1) 0 0; in this case one has

2rr/2=i 67"®1)(rr(9 1). (14) 310 10 DECOMPOSITION OF HOMOTOPY TYPES

Otherwise the order of rT/2 is 2. A complete basis of G(rl'") is given by all elements (13) and all 0-images of basis elements in 7r,(w) ® 71/2 which are not of the form (14). This describes the short exact sequence (12) completely. We now obtain b3 : H3 * G(,1'"). (15) Here L3(w) is a spherical vertex, and hence a basis element in H3, if w = and in this case we get

b3(L3(W))=rQ/2if w=,~,o

On the other hand, R3(w) is a spherical vertex if w or w = e and in these cases we get

r,8/2 if w = b3 (R3(w)) _ 0[r_t ®1] if w =

These formulas define b3. Given b3 we obtain the composite homomorphism

g0(,,'° ® 1): Ho ® 71/2 --+ irl ® 71/2 - G(77w) - cok(b3) (16) where q is the quotient map. As in Definition 8.1.3 we obtain the group G(H2, 71#) by the following push-out diagram A- Ext(H2, cok b3 w)) µ Hom(H2, Ho ® 71/2) push Ext(H2, Ho ® 71/2) v

II °®L, Ext(H2,71/2) ®Ho Hom(G(H2),71/4) ®Ho µ - Hom(H2,71/2) ® Ho

Finally we define the element (3 W in (2) by

/3"'=0(q*13iW)+V(13z)EG(H2,r1 ) (17) where /3i E Ext(H2,G01w)) j62 " E Hom(G(H2),Z/4) (9 Ho are the following elements. Given the canonical direct sum decomposition of G(H2), see Proposition 1.6.5, the element /32 w is the canonical `lift' with

(µ®1)p =b2. (18) 10 DECOMPOSITION OF HOMOTOPY TYPES 311

That is, via (7) the element b2 is a sum of basis elements and /3 is exactly the same sum of the corresponding basis elements in Hom(G(H2), 71/4) ® Ho. Here only Hom(G(71/2),71/4) ® 71/r = 71/4 with r4 is a non-split extension of Hom(Z/2, 71/2) ® 7l/r = 71/2 and in this case the basis element of 71/4 corresponds to the basis element of Z/2. Hence the element (3Z corresponds to all TI-summands in the attaching map f(w) I M(H2, n + 1) in Definition 10.2.6. Next we define 01 similarly by all-summands and e-summands in the attaching map f(w) I M(H2, n + 1). Let 71sT be the subgroup of H2 generated by the basis element s, E B2. Then the inclusion jsr 71/s, c H2 induces (js)*:Ext(H2,G(if ))=Zs,®G(f').

Now we define 01 by the coordinates

s,®rs/2 ifw= 6s= ST-1 h s, ®r,/2 ifw = ... Sle, (j5r)*pi = (19) s3®O(t®1) ifw=tes' ,z=1

S, 1) if w = ,Esc ., z= 1.

This completes the definition of f3'. Using Addendum 2.6.5 and the attaching map f(w) in Definition 10.2.6 we see that R w is actually the boundary invariant of X(w). As in Definition 8.1.5 the element f3 ' yields the extension element (Tr2(w)) and hence we are now able to compute ir2(w)= 7ri+2X(w). Now let (w, (p) be a cyclic word and let H* = H* (w, cp) = H* X(w, gyp) be the homology, with H. = H, = 0. Moreover sinceb,issurjective also or, X(w, (p) = 0. Hence we get the f-sequence

0-+Ho *l/2--*ir2->HZ-Ho ®71/2--0 (20) and the boundary invariant

f3EG(H2,71#)=Ext(H2,Ho *Z/2)ED Horn(H,,Ho ®71/2) (21) with 6 = ( p1, b2). Similarly as above b2 is given by the 71-summands in the attaching map f(w), and 0, is given by the -summands in the attaching map f(w); see Definition 10.2.7 (5), (6). This way we obtain the A3-system S(w, cp) = A'X(w, (p). The isomorphism (21) is natural in H2 hence the extension Ho*Z/2N is given by (7r2) = j *( fi,) where j: ker(b2) c H2 is the inclusion. 312 10 DECOMPOSITION OF HOMOTOPY TYPES

(10.2.14) Steenrod squares for X(w)For finite (n - 1)-connected (n + 3)- dimensional polyhedra X we have the Steenrod squaring operations

(a) Sq2: H2(2) -> H°(2) (b) Sq2: H3(2) --> H1(2) where H;(2)=H"+;(X,71/2) is the homology with coefficients in 71/2. For cohomology groups H'(2) = H"+'(X,71/2) = Hom(Hi(2), Z/2) one has the dual operations

(a)' Sq2: H°(2) -> H3(2) (b)' Sq2: H'(2) - H3(2) which are given by Sq2 = Hom(Sg2,71/2); see (5.2.15). If X = X(w) is given by a word we have a basis of H; = H"+;(X) and the isomorphism

H;(2)=H;®71/29H;_,*Z/2 yields a basis of H;(2); see Definition 10.2.4. In terms of this basis

(a) Sq2: H2®Z/2ED H,*71/2->Ho ®Z/2 is determined in the obvious way by the letters r) in the word w. That is the restriction H2 ® 71/2 -, H° ® Z/2 is defined as b2 in (10.2.13) (7) and the restriction H, * Z/2 --> H° ®1 /2 carries the generator t/2 E H, * Z/2 to r_ 1 ® 1 if w = , ijt , where r-, denotes the spherical vertex if w = 'qt . Similarly

(b) Sq2: H3 ® 1/2 ® H2 *71/2 -> H1 ® 1/2 ® H° *71/2 is determined by the letters 6 in the word w. For X = X(w, (P) we obtain Sq2 by similar formulas as in the definition of the attaching map f(w, (p) in Definition 10.2.7; in fact, in this case we have /3 = ( R1 = Sq2, b2 = Sq2) where /3 isdefinedasin(10.2.13)(21)with Ext(H2, H° *71/2) = Hom(H2 *Z/2, H° *71/2).

(10.2.15) Adem operations for X(w) We first consider w where r and s are powers of two or empty (= 0). Then the Adem operations 10 DECOMPOSITION OF HOMOTOPY TYPES 313

4)00,042,04in Section 8.5 are computed in the following table. r00 so0 X(re' ) X( S) X(51r)

0,(r,s) = (2, >_ 0) 4)' 0 0 St X2, otherwise 0,(r,s)=(>_0,2) O 0 0 X2, otherwise

00 iX2,(r,s)_(0,>0) 0 0, otherwise 0,otherwise f 0,(r,s)=(>_0,2) ,V22, (r, s) = (2, >_ 0) o4 0 t` X4, otherwise 0,otherwise 0, (r, s) = (2, >_ 0) (P2 0 I X22,(r,s)=(>_0,2) Xz,otherwise 0,otherwise

Here X E Hom(Z/n,71/m) is the canonical generator. In the general case we can compute the Adem operations in the same way for X(w) and X(w, P) by using the attaching map f in Definitions 10.2.6 and 10.2.7 and the table above.

(10.2.16) Example For the indecomposable space X(22 e8) we have

46' = 0since ker(Sq') = 0, 0" = 0since ker(Sg22Sq') = 0, ¢° = 0by (10.2.15), 4) =0 by(10.2.15), and only 04 * 0. On the other hand, the indecomposable space X(8 e ` 712) satisfies 0'=O since im(Sg3) = H3(2), 0"=O since im(Sq') = H3(2), .0i = 0by (10.2.15),

442 = 0 by (10.2.15),

and only 0z * 0. These examples show that the classical Z/2-Adem opera- tions0', ¢", 00 do not sufficetoclassify(n - 1)-connected (n + 3)- 314 10 DECOMPOSITION OF HOMOTOPY TYPES dimensional polyhedra. One has to use both operations 04 and 02. This is done in the definition of A3-cohomology systems in Section 8.5.

10.3 The (n - 1)-connected (n + 3)-dimensional polyhedra with cyclic homology groups, n >- 4

We here give an application of the classification theorem 10.2.9. We describe explicitly all indecomposable (n - 1)-connected (n + 3)-dimensional homotopy types X, n >- 4, for which all homology groups Hi X are cyclic, i >- 0. Let H. = (Ho, H1, H2, H3) be a tuple of finitely generated abelian groups with H3 free abelian and let N(H*) be the number of all indecomposable homotopy types X as above, with homology groups H; for i E {0, 1, 2, 3). (10.3.1) TheoremLet n >t 4. The indecomposable (n - 1)-connected (n + 3)- dimensional homotopy types X, for which all homology groups H; (X) are cyclic, are exactly the elementary Moore spaces in A;,, the elementary Chang complexes in (10.2.11), and the spaces X(w) where w is one of the words in the following list. The list describes all w of the theorem ordered by the homology H. s H,,(X(w)). Below we also describe all graphs of such words w; the attaching map for X(w) is obtained by Definitions 10.2.6 and 10.2.7. Let r, t, s be powers of 2. H. = (HoH, H2 H3) N(H*) w with H,K X(w) a H,,

71 71/r 71/t71/s 3 77 s,tSSnryrS> S6r7tS 55r7,t 71/r 71/t71/s 0 3 Tilts, terlr, 71/r 71/t 0 71 2 rftS, ?rr?lt t Z/r 71 71/s 71 1 f fss T 71/r Z 71/s 0 1 12, r=s=2 'lr5,sere and 71/r 0 71/s 71 3, rs-8 fires forrs>>8

f 3,r = s = 2 r Ss, 71r, (n Sr, 1), and 71/r 0 71/s 0 l 4, 8 res fours-8 71/r 0 71 71 'irS 71/r 0 0 71 re

71 71/t71/s 0 'itCCSs, ttSs'i 71 77/t 0 71 7,tf 6 5/ 71 71 71/s 0 ,e r 71 r 0 71/s 0 2 71

7L 0 0 71 1 6 'V 'V 'V 'V 'V

I 4, 316 10 DECOMPOSITION OF HOMOTOPY TYPES

(103.2) Remark Let n >_ 4 and let H. = (Ho, H1, H2, H3) be a tuple of cyclic groups with H3 E {1, 01. Then it is easy to describe, by use of Theorem 10.3.1, all (n - 1)-connected homotopy types X with H; for 0< i< 3 and dim X:5 n + 3. In factall such homotopy types are in a canonical way one-point unions of the indecomposable homotopy types in Theorem 10.3.1. For example for H* _ (1/6,1/2,1/2, 0) there exist exactly nine such homotopy types X which are

M(1/6, n) V M(7L/2, n + 1) v M(1/2, n + 2) M(1/6, n) V X(2 6 2) M(1/3, n) V X(2-q2) V M(1/2, n + 2) M(1/3, n) v X(22) v M(1/2, n + 1)

M(7L/3, n) V X(2, q2) V M(7L/2, n + 1)

M(1/3, n) V X(17''2,1) V M(7L/2, n + 1) M(1/3, n) vX(277 2f 2)

M(1/3, n) vX(262712)

M(1/3, n) V X(262ii2).

It is easy to compute the homotopy groups Trn , Trn+ 1 f 'Tn + 2of these spaces; see (10.2.13). Similarly we see that there are 24 homotopy types X for H* = (71/2,11/2,71/2,1). We leave this as an exercise; compare the list in Baues [HCJ, p. 24.

10.4 The decomposition problem for stable types

A simply connected CW-space X is of finite type if equivalently (a) or (b) are satisfied: (a) all homology groups of X are finitely generated, (b) all homotopy groups of X are finitely generated. Let aR be the full subcategory of Top*/= consisting of CW-spaces X of finite type with trivial homotopy groups zr, X = 0 for i < n and i > n + k. Hence the objects of a are (n - 1)- connected (n + k)-types of finite type which we also call a,-types; compare the notation in (10.1.5). The loop space functor gives us the sequence of homotopy categories (n >_ 2)

(10.4.1) a; =o-a24- an_,ilan 10 DECOMPOSITION OF HOMOTOPY TYPES 317 which is Eckmann-Hilton dual to the k-stem of homotopy categories in (10.1.5). For k + 2 < n the functor fl: an --> an_ is an equivalence of categories, and for k + 2 = n this functor is full and faithful but not representative. We have the Postnikov functor

(10.4.2) Ak+ 1 k Pn+k n --an which is full and representative; see (2.5.2). This is the restriction of the Postnikov functor in (3.4.6) which is compatible with the detecting functors in the classification theorem (3.4.4). In the stable range k + 2 < n we have the commutative diagram

Ak+1+ Ak+1 n n-1

P+kII P,+k-I k an an-1

Moreover we get:

(10.43) LemmaIn the stable range k + 2 < n the Postnikov functor Pn+ k : An+ 1 --> an is an additive functor between additive categories. The biproduct in An+' is the one point of spaces and the biproduct in an is the product of spaces. In particular, one has a canonical isomorphism

Pn+k(X V Y) = Pn+k(X) X Pn+k(Y) forX,YEAn 1

The `decomposition problem for stable types' asks for the complete classification of indecomposable objects in the additive category an, k + 2 < n. There is a relationship between the decomposition problem in An+' and an respectively. For this we recall the following classical `theorem on trees of homotopy types' due to J.H.C. Whitehead [SH]; see also II.§6 in Baues [CHI.

(10.4.4) TheoremLet X, Y be two finite m-dimensional CWcomplexes, m >: 2, and assume X and Y have the same (m - 1)-type, that is Pn,_ ,X = Pm_ ,Y. Then there exist natural numbers A, B such that the one point unions X V V Sm = Y v V S- A B are homotopy equivalent.

The theorem shows that each (m - 1)-type Q determines a connected tree HT(Q, m) which we call the tree of homotopy types of (Q, m). The vertices of 318 10 DECOMPOSITION OF HOMOTOPY TYPES this tree are the homotopy types (X) of finite m-dimensional CW-complexes with P,n_,(X) = Q. The vertex {X) is connected by an edge to the vertex {Y} if Y has the homotopy type of X V S'". The roots of this tree are the homotopy types (X) as above which do not admit a decomposition X= X' V S'.

(10.4.5)Example We describesimply connected 4-dimensional CW- complexes X1, X2 with X,X2 but X, v S4 = X2 v S4. Let

X, = (S2 U. e3), X2 = (S2 U5 e3) U21,, e4 where rl = 'q2 in the Hopf map. Using the detecting functor of Theorem 3.5.6 we see that

A(X,)=(Z-1»I'(71/5)-*O-0,Ho=71/5)

A(X2) = (71 -23 I'(71/5) -> 0 - 0, Ho = 1/5) where F(l/5)=1/5. A homotopy equivalence f: X1 =X2 would give us a commutative diagram

z - 1 Z/5

71 2 Z/5

Here r(f *) has to be a square number (see Corollary 1.2.9) and hence 1'(f) = ± 1 modulo 5. This yields the contradiction, so that X, * X2. On the other hand, the following commutative diagram shows by Theorem 3.5.6 that X1vS4-X2vS4,

71®71 (0' 1) 7L/5 2, 1)1 (5,3® (0, 2) - 71 Z/5

Let Ind(A"') and Ind(al) be the sets of indecomposable homotopy types in An+' and aR respectively. Then Theorem 10.4.4 gives us the following comparison between indecomposable objects in A"' and a'. 10 DECOMPOSITION OF HOMOTOPY TYPES 319

(10.4.6) TheoremLet k + 2 < n. Then the Postnikov functor yields a surjection between sets

Pn+k: (Sn+k+1) Ind(ak) and for Q E Ind(ak) the inverse image P,+ k(Q) is the set of roots in HT (Q, n + k ).

There is only one root in HT(Q, n + k) in case the objects in Ak+' have a unique decomposition. Therefore the theorem of Chang (10.1.8) and the decomposition theorem 10.2.9 show:

(10.4.7) TheoremFor k = 0, 1, 2 the Postnikov functor yields a bijection (n>k+2)

Ind(AR+') _ (Sn+k+1) = Ind(ak).

Hence the decomposition theorem 10.2.9 for A;, actually also solves the decomposition problem in a.. We say that K(A, n)is an elementary Eilenberg-Mac Lane space if A is an elementary cyclic group, that is A = Z or A = 71/p` where p' is a prime power. Using Example 10.1.2 and Proposi- tion 10.1.6 we get for k = 0:

(10.4.8) PropositionLet n >_ 2. The elementary Eilenberg-Mac Lane spaces are the only indecomposable homotopy types in ao and each object in ao has a unique decomposition. Moreover, the Postnikov functor A;, - artcarries an elementary Moore space to the corresponding elementary Eilenberg-Mac Lane space; see Proposition 10.1.6.

For k = 1 the situation is more complicated.

(10.4.9) DefinitionLet p, q be powers of 2 and n >_ 3. Then there is a unique indecomposable space K(71,7L/q, n) in a'with homotopy groups -7rn = Z and irn+ , = 71/q. Moreover there is a unique indecomposable space K(Z/p,7L/q,n) in an with homotopy groups ir, =7L/p and irn+, =7L/q. In fact, K(7L,7L/q, n) = K(77, n) and K(l/p,7L/q, n) = K(77', n) where ri: Z 71/q and 77': i/p -* 71/q are the unique non-trivial stable quadratic functions,thatis,77(1)=q/2 and r7'(1)=q12. We callK(71,7L/q, n) and K(71/p,71/q, n) the elementary Chang types.

In addition to the theorem of Chang (10.1.8) we now get:

(10.4.10) PropositionLet n >_ 3. The elementary Eilenberg-Mac Lane spaces in a;, and the elementary Chang types are the only indecomposable homotopy 320 10 DECOMPOSITION OF HOMOTOPY TYPES types in a;, and each object in an has a unique decomposition. Moreover, the bijection in Theorem 10.4.7,

{Sn+2) Ind(an), is given by the following list where we use the elementary Chang complexes in Theorem 10.1.8. Let p and q be powers of 2.

X PR+]X Sri K(7L,7L/2, n) S"+i K(7L, n + 1) M(7L/p, n) K(7L/p, Z/2, n) M(7L/q, n + 1) K(7L/q, n + 1) X(3t) K(7L, n) X(P'7) K(7L/p, n) X(>7R) K(Z,7L/2q, n) X(pgq) K(7L/p, 7L/2q, n)

Moreover carries an elementary Moore space of odd primes in A;, to the corresponding elementary Eilenberg-Mac Lane space.

10.5 The (n - 1)-connected (n + 2)-types with cyclic homotopy groups, n >- 4

We describe explicitly all indecomposable (n - 1)-connected (n + 2)-types X, n z 4, for which all homotopy groups ir X, aR +, X, vr + 2 X are cyclic. We use the bijection of Theorem 10.4.7 and the computation of homotopy groups iri+2X via A3-systems in (10.2.13). The elementary Eilenberg-Mac Lane spaces and the elementary Chang types in Definition 10.4.9 have cyclic homotopy groups. They correspond to spaces X(w) as follows.

(10.5.1) TheoremThe elementary Eilenbeig-Mac Lane spaces and the elementary Chang types in a., n >- 4, correspond via the bijection

PR+2: Ind(an) = Ind(A.) - (S"+3) to the indecomposable homotopy types in A; described in the following list.

Here Pri+2 carries odd elementary Eilenberg-Mac Lane spaces to the 10 DECOMPOSITION OF HOMOTOPY TYPES 321 corresponding odd elementary Moore spaces. Let r, s, t be powers of 2.

Y Pn+2(Y) E Art K(Z, n + 2) Sn+2 K(7L, n + 1) X( ) K(7L, n) X(2'1) K(7L/s, n + 2) M(7L/s, n + 2) K(7L/t, n + 1) X(tt= ) K(7L/r, n) X(21r6) Sn+1, s=2 K(7L, 7L/s, n + 1) X( C), s=2s'>4 M(7L/t, n + 1), s = 2 K(Z/t,7L/s, n + 1) X(tEs'), s = 2s'-2:4 X(e), t=2 K(7L,7L/t, n) X(art'f ),t = 2t' >_ 4 X(26), t=2,r=2 K(7L/r, 7L/t, n) X(ere), t=2,r>4 X(frflt'f),t=2t'>4

The space P,,+2(Y) describes the (n + 3)-skeleton of Y up to a one-point union of spheres Sn+3, that is

(10.5.2) Y"+3 = V V ,Sn+3 A where A is an appropriate number >_ 0. Part of the list above corresponds to Proposition 10.4.10; see (10.2.11).

Proof of Theorem 10.5.1The right-hand side of the list describes indecomposable objects, hence we have only to show that these objects have the appropriate homotopy groups. This is done by the A3-systems in (10.2.13). For example we have to show that X = X(e'r a,t'4) satisfiesjr,, X = 7L/r, iri+1X=7L/2t', and irn+2X=0. We obtain am+,X by (10.2.13) (9) and we get 1rn+2X = are by the exact sequence

H3=7L(B 7L-b where b3 is surjective by (10.2.13) (15). We leave it to the reader to check the other cases. 322 10 DECOMPOSITION OF HOMOTOPY TYPES

Let ir* = (aro, 7r117r2) be a tuple of finitely generated abelian groups and let NOT *) be the number of all indecomposable homotopy types X with homotopy groups i = 0,1,2 and 1r1(X) = 0 otherwise, n -> 4. (10.5.3) TheoremLet n >- 4. The indecomposable (n - 1)-connected (n + 2)- types X for which all homotopy groups ir;(X) are cyclic, are exactly the elementary Eilenberg-Mac Lane spaces in a2, the elementary Chang types in Theorem 10.5.1, and the spaces Pi+2X(w) where w is one of the words in the following list. The list describes all w of the theorem ordered by the homotopy groups Tr* = 7r* X(w). Below we describe also all graphs of such words w. Let r, t, s > 2 be powers of 2 and for t, s > 4 let 2t' = t and 2s' = s.

7r* =0r0 7r1 7r2) N(7r*) w with 7T * X(w) _ it *

7L 0 7L 11

7L/r 0 7L 17, 6 2s 7L 0 7L/s 11 2r1r frors=2,271rs fors=2s'4 7L/r 0 7L/s 3 2?1rS ,(71Sr, 1) t Si 7L 7L Z/s 1 tt 7L/r 7L Z/s 1 S s7,r S P"+,S",t=s=2 -qt', t=2t'4,s=2 7L 7L/t 7L/s 1 e",t=2,s=2s'>4 t=2t'>4,s=2s'>4 7L/t 7L/s s' s61t, 2 6rntr , >; t>-4s>_4 witht = 2t', s = 2s' 7L /t 7L/r 7L/2 1 4rit', t = 2t' t > 4 Z/s 71/2 2 s>-4 s s = Z/s 71/2 71/2 2 and s>4 s

and 71/2 7L/2 2 rS

71/2 71/2 71/2 1 26 10 DECOMPOSITION OF HOMOTOPY TYPES 323

For all tuples of cyclic groups ir,, = (ire, Try, 70, aro * 0,7T2 # 0 which are not in the list we have N(Tr, ) = 0. If ir0 = 0 or 7r2-0 we use the list in Theorem 10.5.1. All words in the list are special words, except the word (1r' 1) which is a special cyclic word associated with the identity automorphism 1 of Z/2. We now show the list of graphs:

n nr 2sn

E' 4117, S '7r4 of

lgrnfl

v 24s M(72,n)

re r 324 10 DECOMPOSITION OF HOMOTOPY TYPES

Here s and t satisfy the conditions described in the list of words above.

(10.5.4) Remark Let n >_ 4 and let ir* _ (Ir0, 7r Ir2) be a tuple of cyclic groups. Then it is easy to describe, by use of Theorem 10.5.3, all homotopy types X with 7T,, j (X) = 7ri for i = 0, 1, 2 and 7rjX = 0 for j < n and j >n + 2. In fact all such homotopy types are in a canonical way products of the indecomposable homotopy types in Theorem 10.5.3. For example for ?r * = (7/6,1/2,1/2) there exist exactly seven such homotopy types X, which are K(1/6, n) x K(1/2, n + 1) x K(1/2, n + 2) K(1/6, n) x K(1/2,1/2, n + 1) K(1/3, n) x K(1/2,1/2, n) x K(1/2, n + 1) K(1/3, n) x K(1/2, n + 1) x

K(1/3, n) x K(1/2, n + 1) x Pi+2X(712 K(1/3, n) x K(1/2, n + 1) x Pi+2X(r12C2,1) K(1/3, n) x

It is clear how to compute the homology H,,, and Hi+2 of these spaces and, in fact, we can easily describe the A3-system of these spaces. We leave it to the reader to consider other cases, for example for ir,, = (14,7110,1) there exist exactly three homotopy types X with 7r * = IT * X.

Proof of Theorem 10.5.3Let 7r * = (7r0, ir,, ir2) be a tuple of cyclic groups 0, 1, or 1/2k and assume 7r0 * 0 and vr2 * 0. We want to describe all indecomposable X in A;, with 7ri for i = 0, 1, 2. We clearly have 7r0 = H0 and the exact sequence

H3-G(7'")-->TT2-->H2b2H00 7112'7Tr,-->H,-->0 (1) where H0 ® Z/2 =1/2 and we have the extension

7T , ®1/2 --s G(714') -. H0 * 1/2. (2)

Moreover 7r2is determined by /3 as in (10.2.13).

First case, ar, = 0 Then also H, = 0 and b2 * 0; moreover H2 is cyclic or H2 = cyclic ®71/2 since ''r2is cyclic. We have G(ri') = H0 *Z/2 since IT, = 0. Hence H3 =1 or H3 = 0. The only special cyclic word with these properties is X = X(,11,,1) 10 DECOMPOSITION OF HOMOTOPY TYPES 325 satisfying aro = 71/r, are = Z/s; see (10.2.13) (21) for the computation of are. An e-word w for X = X(w) is not possible since b20, also a central word is not possible since H, = 0. Hence it suffices to consider basic words with the homological properties above. The basic words w are

71, s 77, ir,sr, 17r6, qrS 2,1r S, qlS 2and 2T1r5 S. (3)

We obtain are = by the remark following (10.2.13) (19). For w in (3) we get (2s' = s) are = 71,71/s',1 ® 71/2,Z/s' ® 71/2, 71, 71 ®71/4,71/s', l/s' ®71/4 and71/2srespectively.

Hence the cyclic cases of IT, are those in the list with ar, = 0. For this we replace s' resp. 2s by s. This completes the first case.

Second case, -rr,0 and b2 * 0 Since b,0 we have -q' = 0 and Ir, = H, * 0 is cyclic and G(77'") = ir, Z/2 ® Ho *71/2. If H, = 11/t then w has to be a central word, but since b2 * 0 and Ho cyclic, this is not possible. If H, = 1, w has to be a basic word of the form 'r7 for Ho = 7, or one of e'a7, and 6'17,and a1, 6 2 and 6 271,6' for Ho = 71/r. However for Ho = 71/r we have H30 since G(77w) is not cyclic. Hence only e71,6 and 6'a7 remain. For X( q) we have or, = H, = Z = Ho. are is a non-trivial extension and hence or, = 71/s. For X('77,) we have ar, = H, = Z and Ho = 71/r and oreis a non-trivial extension with ar, = 71/s. This completes the second case.

Third case, rr I * 0 and b2 = 0 and H, * 0 Then H2 is cyclic and we see that ar,is a non-trivial extension of H, by 71/2 and hence w has to be a central word with H, = 71/t. Moreover for H, * 0 and Ho = 71/r we have H3 * 0 since G(if) is not cyclic. Hence we get the possible words o7t, aitC, qt;r', t=,ait, e,,7te, ,a7tC', or '6,-7tC. Here r7tC and S,olt5 appear in the list of Theorem 10.5.1 with are = 0. For rit we get ar, = 71/21 and ore = Z/2. For 1716' we get Ir, = 71/21 and are = i/2s.For ,ait we get or, = i/2t and ore = 71/2. For 6r7jt4' we get Ir, = 71/2t and ore = 71/2s. For6,o71 we get ar, = l/2t and or, = 71/2s. This completesthe third case.

Fourth case, o:I * 0 and b2 = 0 and H, = 0 Then H2 is cyclic and 17": 71/2 = ar,is an isomorphism. Now w has to be a basic word of the form,f 6 ' (since b2 = 0) or an ---word of the form e, , e, 326 10 DECOMPOSITION OF HOMOTOPY TYPES e-s, res (rs > 8), C,_- (r We compute?r2(w)=rri+2X(w) as follows: 7r2(, 6) = 0 for r = 2 and = 11/2 for r >_ 4, 71/4s for r = 2 and non-cyclic otherwise, 1r2(e) = 0,-rr,(, e) = 71/2, zr2(es) = l/2s, 1r2(res)=71/2®71/2s, 7r,Qre)=0 for r * 2. 1r2() =71/2s, 7r,(5 e)= 71/2s. This completes the fourth case.

Final case We finally have to consider indecomposable spaces X in A;, which are not of the form X(w), X(w, cp) and for which 0, iri+2X* 0 are cyclic. The only possibilities for X are the elementary Moore spaces S' and M(Z/2, n).

(10.5.5) Remark on k-invariantsLet n > 4 and let a* _ Oro, 1T 1, 7T2) be a tuple of abelian groups. The classical approach to classify homotopy types Y with homotopy groups ir;for i = 0, 19 2 and 0 for j < n, j > n + 2 uses the Postnikov tower and the k-invariants of Y. For this we first choose a homomorphism

71: Tro®Z/2-7T, and then we choose an element

k E H"+3(K(i, n), ire)

Then there is a unique homotopy type Y(rt, k) with k-invariants -1 and k respectively. We have the split short exact sequence

Ext(Hi+,K(i,n),a,) N H'n+3(K(rt,n),IT2) -_3P

Ext(ker(i1), ar,) Hom(G(q), a2) where G(q) is determined by 77 as in Definition 8.1.3 and ker(-7) c iro ® 71/2. The split exact sequence, however, does not show us how the group of homotopy equivalences'(K67, n)) acts on the direct sum Ext(ker(rl), ire) Hom(G(71), 7r, ). This action is needed if we want to classify the homotopy types Y above since for a c= 9'(K67, n)) the spaces Y(i7, k) and Y(q, a*k) are homotopy equivalent. Using the `theorem on k-invariants' (2.5.10) we have relations between the Postnikov invariant k and the exact sequence

_q) H3 - G( * 7r, -, H2 -4 7ro 0 71/2 a, -> H, - 0 10 DECOMPOSITION OF HOMOTOPY TYPES 327 which is Whitehead's F-sequence for the (n + 2)-skeleton X of Y(ri, k). Here k * = µ(k) is given by µ above and the element

kt = A-'q* (k) E Ext(ker(q), cok(k*))

determines the extension

cok(k *) N H2 -s ker(b) given by the exact sequences. We can apply these facts to get some hold on the k-invariant of the spaces described in the list of Theorem 10.5.3. For example for

w = t = 2t', s = 2s', we have 'ro = 71/r, ir, = l/t, 7r2 = 71/s and 71 = if : l/2 `-a71/t is the inclusion. Hence µ above is an isomorphism and the k-invariant is k=µ-'(k) EH"+3(K(rr,n),jr,) = Hom(G(rl),?r2) kG(rr)=ir, ®71/2®77)*71/2=Z/2ED 71/2-X71/s=ir2.

Here k* is trivial on 770 *71/2 =1/2 and is the inclusion s': 7/2 71/s on 7r, ® 1/2 = 1/2. This follows from the A3-system associated with w in (10.2.13). For w' we obtain 77 = 17" as above and k = µ-'(k * ) where k * is trivial on 7r® (9 1/2 and the inclusion on 770 *71/2. By Theorem 10.5.3 the spaces Y=Pi+2X(w) and Y' =Pi+2X(w') are the only indecomposable homotopy types which realize 7r* _ (71/r,Z/t,l/s), t, s >_ 4. The X(w) and X(w') correspond to the (n + 3)-skeleton of Y and Y' respectively; see (10.5.2).

10.6 Example: the truncated real projective spaces LF P+ 4/If8 P

The real projective space RP is a CW-complex with n-skeleton RP,, and with exactly one cell in each dimension. Hence the quotient spaces

(10.6.1) P3 = O I are (n - 1)-connected (n + 3)-dimensional spaces. We here show how these spaces fit into the classification. For n >_ 4 the spaces P3 are stable; moreover since the homology groups of P3 are cyclic, the 2-connected 6-dimensional spacesp,3, y.P,, ,2P, with P,= RP4 are determined by their stabilization; 328 10 DECOMPOSITION OF HOMOTOPY TYPES see Corollary 9.1.16. Hence it suffices to consider the stabilization ofp,,3, n > 1, and the simply connected 5-dimensional spaces I(IRP4) and RP5/S1 where S 1 = U8 P1. We say that two finite CW-complexes X, Y are stably (E')-equivalent if there exists a homotopy equivalence Y .'X = I ' Y for some n,m - Owith In - ml=r.

(10.6.2) Theorem One has stable equivalences, n >_ 1,

X(2 6 2) forn = 1(4) for n = 2(4) 3- P" X(2172 for n = 3(4) S" V Sn +3 v M(Z/2, n + 1)forn = 0(4).

Hence the graphs of these stable spaces are (k > 0)

3 3 P4k+1 P4k+2 P4k+3 P4k where P4k with k >_ 1 is a one-point union of Moore spaces. For the stable homotopy groups it X= "M .kXwe get the list below which we derive from the equivalence in Theorem 10.6.2. Let

(10.6.3) P = 1ESP /R P,, -I.

Then we have for k:!5; 2 the isomorphism i +k(P,) =+k(P3) and this group is computed in the following table.

(10.6.4)

n>_1 7r's(P,-) + 1(P n=4k+1 71/2 71/2 71/8 n=4k+2 71 Z/4 0 n = 4k + 3 71/2 0 Z/2 n=4k 71 1/2 ® 1/2 Z/2 ® 1/2

Proof of (10.6.4)The case n = 4k + I is considered in Example 8.1.11 or in Theorem 10.5.3. The case n = 4k + 2 is obtained by Theorem 10.5.1 since Pi+2X(r)t) = K(71,Z/2t, n), t>2. 2.Moreover for n = 4k + 3 see again Theorem 10.5.3. 10 DECOMPOSITION OF HOMOTOPY TYPES 329

P r o o f o f T h e o r e m 10.6.2 Let U E H2(!F P, ,71/2)Z/2 be the generator. Then the n-fold cup product u" e H"(IRP ,71/2) = Z/2 is a generator and the Steenrod square Sq' satisfies

Sq`(u")=(n)un+1 (1) compare 2.4 in Steenrod and Epstein [CO]. The formula also yields the action of Sq2 on H*(P3,Z/2). Now one readily checks that the equivalences in Theorem 10.6.2 correspond to isomorphisms of homology groups compatible with the action of Sq2. The classification of homotopy types with cyclic homology groups in Theorem 10.3.1 and (10.2.14) now shows that the first three equivalences hold. The case P 4k is more complicated since Sq 2 acts trivially. Hence we get either an equivalence as in the theorem or

P4k M(71/2,4k+1)VX(e). (2)

Considering Pk as a Thom space shows readily that (2) does not hold. We also refer to Davis and Mahowald [CS] where all truncated spaces Pk are classified up to stable equivalence.

We now consider the unstable spaces 1:R P4 and IF P5/S' which are simply connected and 5-dimensional.

(10.6.5) TheoremThe space I R P4 is the mapping cone of

(1) 2+ y22: P- - PZ .

Moreover the space 111 PS/S' is the mapping cone of

(13713 + [13,12],213 +12'112): S4 V S3 - S3 V S2. (2)

Here i3 and i2 are the inclusions of S3 and S22 respectively and [i3,i,] is the Whitehead product. In (1) we use the generators 6z , y; defined in (11.5.16) with (y2) = 0. We point out that II P4 = T,(4) is one of the Brown-Gitler spaces in Goerss, Lannes, and Morel [VW].

Proof of Theorem 10.6.5 We know that II1 P4 is stably X(2 f 22), therefore the attaching map has to be f= fi + Sy2; see (11.5.9), with S E (+1, -1). In Baues [CHI IV.A.11 we have seen that Y. P3is the mapping cone of (2712, -2): S3 V S2 - S2. This shows that S = +1. On the other hand, we proved in Baues [CH] IV.A.9 that Fl P4/S' is the mapping cone of

12712+213: S3-S2 VS3=FlP3/S'. 330 10 DECOMPOSITION OF HOMOTOPY TYPES

We therefore get the following F-sequence for RP.,/S':

H5 - F4 - zr4-* H4 - FH2 "Ir3 ---> H3 -+ 0

II II it II II II II 2 71 1/2 (B 71/2 11/2 0 71 71 Z/2

Since H2 = 71 we get

F4 = F4(l P5/S') = 17226q) = 71/2(i3r13) ® 71/2[i3"21 compare Section 11.3. Now b5: H5 - F4 is non-trivial since 48 P5/S' is stably X(7120). In fact this implies b5(1) = i3713 + 5[i3, i2] with o E (1, -11. Since there is a non-trivial cup product (71/2-coefficients) H3(2) ® H2(2) -> H5(2) we see that S = +1. The boundary invariant f3is trivial since H4 = 0 and hence the homotopy type of I}BP5/S' is determined by the exact sequence above. This yields the required homotopy equivalence for ll P5/S'.

(10.6.6) CorollaryWe obtain the homotopy groups 7r3(YIRP4) = 71/2 and 7r4(E18P4) = Z/4 and 7r3(1 P5/S') = 71 and ir4(RP5/S') = Z/2.

Proof The F-sequence for 11 P5/S' is described in the proof of Theorem 10.6.5. We now consider the F-sequence for 7fl P4:

H5 --F4 ->Ir4 -_L H4 - FH2 - 'r3--> H3 -* 0

II II II II II II II 0 Z1/4 = Z/4 Z/21/4 - Z/2 0

Here b4 is non-trivial and hence F4 = H4 is an isomorphism. Moreover we have in (11.3.7) the isomorphism n.: F2 (H2) = 172 1(77) =1 /2 so that a,,: Z/4 = ir4 M(H2, 2) = F4 is an isomorphism

(10.6.7) Corollary[Y-RP4,S2] =1/2. We leave this as an exercise, use (11.5.25).

10.7 The stable equivalence classes of 4-dimensional polyhedra and simply connected 5-dimensional polyhedra In the decomposition theorem 10.2.9 we determine for n > 4 the set 1 of homotopy types of finite (n - 1)-connected (n + 3)-dimensional polyhedra. The homotopy types (I"-'X), where X is a finite connected 4-dimensional 10 DECOMPOSITION OF HOMOTOPY TYPES 331 polyhedron, forma subset X(4) c.3. Moreover the homotopy types {fin-'Y}, where Y is a finite 1-connected 5-dimensional polyhedron, form a subset X(5) with (10.7.1) 3r (4) C X (5) c X. We now describe these subsets explicitly in terms of the spaces X(w) and X(w, cp) used in the decomposition theorem.

(10.7.2) DefinitionLet r, t, s >_ 2 be powers of 2. The 4-dimensional words are , C' with r >_ s, and t6 s, te,s, . The corresponding graphs are

The next result describes the set 1(4) in terms of 4-dimensional words.

(10.7.3) TheoremLet n >_ 4 and let X be a finite connected 4-dimensional CW-complex. Then there is a decomposition (unique up to permutation)

In - IX _ XI V . V Xk.

Here the complexes Xi for i = 1, ... , k are elementary Moore spaces or spaces X(w) where w is a 4-dimensional word. Moreover for each 4-dimensional word w there is a finite connected 4-dimensional complexA(w) with En-IA(w) = X(w).

Compare (V.A.4) in Baues [CH] where we call the spaces A(w) 'elementary cup square spaces'.

(10.7.4) DefinitionLet w be a special word as defined in Definition 10.2.1. We say that w is a 5-dimensional special word if w satisfies the following properties (1) and (2):

(1) w * 71s and w 0 il (2) for each subword of the form ruls or hlr of w (that is w = rhs or w = Sflr ... ) we have 2r

Moreover a special cyclic word (w, 'p) in Definition 10.2.1 is 5-dimensional if w satisfies (2). Now the next result describes the set 1(5) in (10.7.1) in terms of such 5-dimensional words. 332 10 DECOMPOSITION OF HOMOTOPY TYPES

(10.7.5) TheoremLet n > 4 and let X be a finite 1-connected 5-dimensional CW-complex. Then there is a decomposition (unique up to permutation)

In-2X=XIV ... VXk.

Here the complexes Xi for 1 = ...1, ,k are elementary Moore spaces or spaces X(w) and X(w, gyp) where w and (w, gyp) are 5-dimensional special words and 5-dimensional special cyclic words respectively. Moreover for each 5-dimensional special word w and for each 5-dimensional special cyclic word (w, go) there exist finite 1-connected 5-dimensional CW-complexes B(w) and B(w, gp) respectively such that 1n-2B(w)=X(w) and In-2 B(w, tp) = X(w, (p).

ProofThe existence of B(w) and B(w, (p) follows from Theorem 11.6.6 below by desuspension of the attaching map f(w) and f(w, (p) in Definitions 10.2.6 and 10.2.7. Now let X be a finite 1-connected 5-dimensional CW- complex. Then the decomposition theorem 10.2.9 yields an equivalence yn 2X- X, V V Xk where the Xi are elementary Moore spaces or X(w) or X(w, go). We have to show that w satisfies (1) and (2) in Definition 10.7.4. For this we use the Pontijagin square p for which we have the following natural commutative diagram of homomorphisms; see (1.5.3) in Baues [CHI and J.H.C. Whitehead [CE].

FH2(X,A) P H4(X,FA)

I- I H22(X, A) ® 71/2 H4(X, A (9 Z/2)

II II H2(X,7L/2) ®A 592®A H4(X,7L/2) ®A

Here A is any abelian group of coefficients. Now the commutativity of this diagram implies that w has to satisfy conditions (1) and (2) in Definition 10.7.4. In fact, assume w = ,,7s . Then r and s correspond to basis elements (see Definition 10.2.4)

e,EHn(yn-2X), e5EHn+2(jn-2X) of order r and s respectively. For n = 2 we obtain by e, the dual basis element e* E H2(X, A) and es E H4(X) determines a direct summand Hom(Z/s, FA) c H4(X, FA) with A = 7L/r, F(A) = 7L/2r. Now by (10.2.14) W = ... ,1]s implies that (Sq2 ®A)o,(e*) # 0 and therefore the coordinate cp E Hom(7L/s, FA) of p(ye*) E H4(X, FA) satisfies o-(go) # 0. This implies s >_ 2r. Here y and o are the functions in (1.2.1), (1.2.2). In a similar way we see that w satisfies (1) in (10.7.4). 11 HOMOTOPY GROUPS IN DIMENSION 4

The computation of the homotopy groups irn X, n >_ 1, of a connected space X is a fundamental problem of algebraic topology. It is well known how to determine the fundamental group 7T, X in terms of the attaching maps of 2-cells in X. For n >_ 2 we may assume that X is simply connected since zr X coincides with the corresponding homotopy group of the universal cover of X. For a simply connected space the Hurewicz theorem shows that 7r2 X is isomorphic to the homology H2 X and that the Hurewicz homomorphism 7r3X -> H3 X is surjective. J.H.C. Whitehead considered the homotopy group 7r3(X) and showed that one has an exact sequence

H4 X b`-> I'(H2 X) ----i ?r3X -> H3 X -O. For this he computed the group r3(X) by the natural formula F3(X) _ I'(H2 X ). The corresponding computation of or4(X ),however, was not achieved in the literature. In this chapter we compute I'4X = F4(rl) in terms of the homomorphism rl: r(H2 X) --> ir3 X so that ir4 X is now embedded in the exact sequence

b4 HSX which extends the sequence of J.H.C. Whitehead above. The formula r4(X) = r4(ri) relies on the computation of the homotopy group IT4 M(A, 2) of a Moore space of degree 2. Here A is an arbitrary abelian group. The results in this chapter are crucial for the classification of simply connected 5- dimensional homotopy types in Chapter 12 below where we also study the functorial properties of r4(71).

11.1 On 7r4(M(A, 2))

Let A be an abelian group. In this section we embed the homotopy group ir4M(A,2) in a natural short exact sequence. For this we need the following algebraic functors which carry abelian groups to abelian groups. First we recall from Definition 6.2.7 the definition of the r-torsion FT(A) of A. If dA:A, -A0 is a short free resolution of A, then we have (11.1.1) rT(A) = kernel(5,)/image(5,) with s,-r(A0) A, ®A, r(A,) ®A, ®A0 334 11 HOMOTOPY GROUPS IN DIMENSION 4 given by 61 _ (F(dA), [dA,1]) and 52 =(11,11, -1 ®dA). Let I'* (dA) be the chain complex given by (82, 6,) and which is concentrated in degree 0, 1, 2; see Definition 6.2.5.

(11.1.2) Proposition We have FT(A) =A *Z/2 ifA is cyclic, and FT(AED B)=FT(A)a) FT(B)

These formulas easily allow the computation of TT(A) for all direct sums of cyclic groups A since I'T is compatible with direct limits.

Proof of Proposition 11.1.2:Clearly FT(7L) = 0. For A= 7L/n let dA = n: 71 - 7L so that T * dA is 7L-7Lgi-4Z with 62(1) = (2, 0) - (0, n) and 6,(x, y) = n2x + 2ny. This shows FT(7L/n) = i/n * i/2. Next we prove thecross-effect formula FT(A I B) =A * B. For this we consider the cross-effectF,, (d,I dB) in the chain complex r* (dA ® dB). We readily see that F* (dA I dB) is given by

d' A,®B,®(A,(&B0®B0®A,).9, A,®B,ED B, (&A, A0®B0.

We map this chain complex to the chain complex dA ® B concentrated in degree 0 and 1. Let q: B0 -> B be the map of the resolution of B. Then we map F*(d,. I dB) to dA ®B by l 0q on A0 ®B0 = Fo(dA I d,) and by (0,1 ® q, 0) on F,(dA I dB). This chain map h: I'* (dA I dB) -+ d® ® B induces isomorphisms in homology in degree 0 and 1. Hence FT(AIB)=H,1'*(dAIdB)=H,(dA®B)=A*B.

Compare the more general proof in Baues [QF] 7.3. O

Whitehead's F-functor is endowed with natural homomorphisms, see Section 1.2,

(11.1.3) A®71/2- FA-LA®A given by o, (y(a)) = a 0 1 and H(y(a)) = a ® a for a EA. Similarly we obtain natural homomorphisms

(11.1.4) A*Z/2 °--FTA- H A*A as follows. We define chain maps

h dA ®7L/2 F* dA dA ® A 11 HOMOTOPY GROUPS IN DIMENSION 4 335 by Qo = o,, ho = (1 ® q)H, o,, = (o,, 0), and h, = (0,1 ® q). These chain maps induce (11.1.3) and (11.1.4) in homology. We have for a, b, d E A with ha = hb = 0 and 2d = 0 the formulas

HTh(a, b) = Th,(a, b) - Th,(b, a) Hy(d)=TZ(d,d) crrh(a, b) = 0 uy(d) =d.

Here y(d), Th(a, b) E I'T(A) are the generators in the proof of Theorem 6.2.9. We may assume that h is a power of a prime. Next we need the following notation on Lie algebras.

(11.1.5) DefinitionLet T(A, 1) be the free graded tensor algebra generated by the abelian group A where A is concentrated in degree 1. Thus T(A, 1) is a graded 71-module for which

T(A,1)" =A (& ... ®A =A®" (1) is the n-fold tensor product of A. We define the structure of a graded Lie algebra on T(A, 1) by

[x,y] =xy- (2) for x, y E T(A, 1). Let L(A, 1) be the sub-Lie algebra generated by A in T(A, 1) and let L"(A,1) = L(A, 1) n A®". Clearly, L"(A,1) is a functor which carries abelian groups to abelian groups. For L3(A, 1) we have a further characterization as follows: consider the triple Lie bracket homomorphism

[[1,1],1]: A ®A ®A A ®A ®A (3) which carriesa®b®cto[[a,b],c]=(a®b+b(ga)®c-c®(a®b+ b(9 a).

Lemma The image of [[1,1],1] is L3(A,1) and the kernel of [[1,1],1] is the subgroup W3 in A ® A ® A which is generated by the following elements (a) a®b®c+c®a®b+b®c®a (b) a®b®c-b®a®c (c) a ®a ®a. 336 11 HOMOTOPY GROUPS IN DIMENSION 4

Therefore we have the natural isomorphisms

L3(A, 1) _ [[A, A], A] =A ®A ®A/W3. (4)

Here [[A, A], A] denotes the image of [[1,1],1] above. Moreover we derive from Whitehead's quadratic functor IF the following functor F2: Ab -p Ab

(11.1.6) DefinitionLet F2(A) be the quotient group

r 22(A) = (T(A) ® Z/2 ®1'(A) (&A)/M(A) where M(A) is the subgroup generated by the elements

y(x) ®x (1) [x,y]®1+y(x)®y+[y,x]®x (2) withx, y E A. A homomorphismgyp: A --> B induces F2(A) -- F2(B) by F((p) ® 7/2 ® F(cp) ®cp. The relation (2) implies that

[x,x']®y+[y,x]®x'+[y,x']®xEM(A). (3) For this replace x in (2) by x + x' E A.

(11.1.7) Lemma There is a natural isomorphism

F22 (A)= f(A) ® 71/2 9 L3(A,1) which carries u ® 1 with u E F(A) to the equivalence class of u ® 1 in r22(A) and which carries [[x, y], z] E L3(A,1) to the equivalence class of [x, y] ® z.

We define a homomorphism

(11.1.8) A: 1722 (A) -> 7r4M(A,2) as follows. Using the identification A = 7r2 M(A, 2) and r(A) = ar3 M(A, 2) the function A carries u ® 1 E r(A) ® 7/2 to the composite u'13 where

773:S4 - S3 is the Hopf map. Moreover A maps [[x, y], z] E L3(A,1) to the triple Whitehead product [[x, y], z] E ir4M(A,2) and maps u ®x E F(A) ®A to the Whitehead product [u, x] E 7r4 M(A, 2). The relation (1), (2), and (3) in Definition 11.1.6 correspond to the following classical formulas so that A is well defined:

(1) the equation [173, 62] = 0 where 62 E ir2S2 is the generator; (2) the Barcus-Barratt formula which for a, b E are X yields the equation [Q113, b] = [a, b]174 - [[b, a], a]; 11 HOMOTOPY GROUPS IN DIMENSION 4 337

(3) the Jacobi identity for Whitehead products which for a, b, c E 7r2 X yields [[a,b],c]+[[c,a],b]+[[b,c},a]=0. (11.1.9) TheoremThere is a natural short exact sequence (A E Ab)

I2 (A)>A) 1r4M(A,2)A rT(A). Moreover AL3(A,1) is a direct summand of 7r4M(A, 2), unnaturally. In the proof of Theorem 11.1.9 we need the following notation for mapping cones. Let f: X1 -->X0 be a map in Top* and let Cf be the mapping cone of f. Hence Cf is the push-out CX1 -L Cf

(11.1.10) U U if X1 f'Xo where CX1 is the cone of X1. Let 7r (X1 V X0), = kernel(0,1).: 7r (X1 V Xo) - 7r (Xo ). Then we obtain the following commutative diagram with exact rows

7T,, ,1(CX1 V Xo, X1 V Xo) 7r (X1 V Xo)2 1(rf,1). 1(f, l). d 7rR+1Cf- 7f»+1(Cf,X0) 7r,(Cf) The bottom row is the exact homotopy sequence of the pair (Cf, X0). We call

Ef= (7rf, 1). d-' :7T,, (XI V X0)2 -*nrn+1(Cf, XO) the functional suspension. The operator EE is part of the EHP-sequence in Section A.6 in the appendix; see also Baues [AH] and Baues [OT].

Proof of Theorem 11.1.9Compare the proof of Theorem 6.15.13. Let f: X1 =M(A1,2) -X0 =M(Ao,2) (1) be a map which induces the resolution dA: Al -A 0, that is dA = H2(f ). Then the mapping cone of f is the Moore space Cf = M(A, 2). We now obtain the commutative diagram 7r3(X1 VX0)2 = F(A1) ®A1 ®Ao

(2) EEl 1b, 7r4M(A,2) -- 7r4(Cf, Xo) - 7r3Xo = I'A0 d 338 11 HOMOTOPY GROUPS IN DIMENSION 4 whereS,isthe operator in (11.1.1). The EHP sequence shows that kernel(EE) = image(82). Hence we obtain a well-defined homomorphism.

kernel(S ) (Ef)j: 7r4M(A,2) - I T(A) = (3) image(52)

This homomorphism is surjective since the EHP sequence shows that EE is surjective. Moreover

kernel( µ) = kernel(j) = image(ir4Xo --* ir4M(A,2)). (4)

We have to show that 0 in (11.1.8) induces an isomorphism

A: fz(A)-->kernel( µ). (5)

For this we consider the diagram

?r4(X, vXo)2 (f,1).

or5(Cf, Xo) ---ir4(Xo) - 7T4M(A,2) a where the EHP sequence shows that EE is surjective. Hence we get

kernel( A) = Tr4(Xo)/(f,1) * ir4(X, v X0)2. (6)

Now we can use the Hilton-Milnor formula for ir4(Xo) and 1r4(X, v Xo) which shows by (6) that (5) is an isomorphism; we omit the somewhat tedious computations. One readily checks this way that 0 in (5) is surjective. For this injectivity of A in (5) we can use Lemma 11.1.7 and the following commutative diagram where y3 is the James-Hopf invariant.

r(A) ®Z/2 7r4M(A,2) 4 W L3(A,1)

1i 7r4lM,, AM,, AM,, = ®3A

Here i is the inclusion and w is the triple Whitehead product. The diagram implies that w is injective and that the injectivity of A in (5) follows from the injectivity of 17*. In 11.1.17 we show that OL3(A,1) is always a direct summand. 0 11 HOMOTOPY GROUPS IN DIMENSION 4 339

We consider the (n - 2)-fold suspension operator y" 2 which for n z 4 is part of the following commutative diagram

r2(A) >->Ir4M(A,2) 17T(A)

(11.1.11) lif 1- A OZ/2 - 7T,,-2M(A,n) -A *Z/2

Here o, on TT(A) is defined in (11.1.4) and ii on f2 (A) is defined by (11.1.3) with 5; (FA ®A) = 0. Both maps o, and Q are surjective so that also 2 is surjective. If A is cyclic then o and Q are isomorphisms. Hence we get

(11.1.12) PropositionFor a cyclic group A one has the isomorphism (n >_ 4)

1n-2: 7r4M(A, 2)= ir,+2M(A, n).

This also shows that the sequence in Theorem 11.1.9 in general does not split since we know the group it"+, M(A, n) = G(A) by Theorem 8.2.5.

We introduce two new functors

(11.1.13) ir4, 7r4 : M2 - Ab given by the natural quotient groups

ir4M(A,2) = ir4M(A,2)/Al'(A) ® 71/2, (1) ir4'M(A,2) = 1r4M(A,2)/OL3(A,1). (2) Here we use Lemma 11.1.7 and Theorem 11.1.9. The exact sequence in Theorem 11.1.9 induces the natural short exact sequences

A-> L3(A, 1) a4M(A,2) - FT(A) (3)

1'(A) ®71/2 >s-> Tr4M(A,2)-`'1'T(A). (4)

We shall prove that the extension (3) is split for all abelian groups A. Using (3) and (4) we obtain the natural pull-back diagram

ir4M(A,2)-ir4M(A,2) q1 pull (5) ir4M(A,2) 1'T(A) 340 11 HOMOTOPY GROUPS IN DIMENSION 4 in which q denotes the quotient map. This pull-back diagram shows that the functor IT4 on M2 is completely determined by the functors i4 and 7r4 on M2. The functor i4 has the following natural interpretation. Let (I M(A, 2) be the universal cover of the loop space 1 Z M(A, 2). Then one has the natural Hurewicz homomorphism

ir4M(A,2) 7r3f1:M(A,2) = ir3,ilM(A,2)-_,H3SlM(A,2). (6)

Here the third homology H3 of A M(A, 2) is isomorphic to 7r4 M(A, 2). In fact, there is an isomorphism

7r4M(A,2)H3,f1M(A,2) (7) induced by hq-' where q is the quotient map in (5).

Proof of (7)The map h in (6) is embedded in Whitehead's exact sequence of X=IZM(A,2)

FH2X----a 7r3X=4H2X-0.

Here H2 X = 7r3M(A,2) = IF(A) and j coincides with the composite j: FA- F(A)®Z/2N7r4M(A,2)=ar3X.

Hence the kernel of h is f(A) 0 1/2 and therefore the isomorphism (7) is well defined.

We now compute the group ir4M(A,2) for any abelian group A. For the Moore space M(A,2) = IMA we have the James-Hopf invariant

y3: ar4M(A,2) - zr4IMA AMA AMA =®3A which satisfies y3OF(A) ® 1/2 = 0. Hence y3 induces the following commutative diagram in Ab with short exact rows.

L3(A,1)N ir4M(A,2)--FT(A) (11.1.14) I Y3 I Y3 11 L3(A,1) - `0 ®3A - Q - ®3A/L3(A,1)

Compare the proof of Theorem 11.1.9. This diagram is a pull-back diagram which determines the group a4 M(A, 2) via the operator 'y3. This operator is 11 HOMOTOPY GROUPS IN DIMENSION 4 341 not natural in A. For the computation of y3 we need the James-Hopf invariant y2. Yz [IP2,IMA] [Y. P,,IMAAMA] (11.1.15) 1A II A D Hom(Z/2, A) Ext(l/2, 0 2A)«- ®'-A

For dEA with 2d = 0 let y2(d) E ®2A be an element which represents y,(d) with J C= µ-'(d). As usual we identify an element a EA with ha = 0 with an element a (=- Hom(Z/h, A) carrying 1 E i/h to a.

(11.1.16) TheoremLet h be a power of a prime and let a, b, d c- A with ha = hb = 0 and 2d = 0. Then we have y3(rh(a, b)) = 0 if h is odd and we get in ®3A/L3(A,1) the formulas

y3(rh(a, b)) _ (h/2)[a + b, a ®b]for h even, y3(y(d)) = [d, y2(d)]. Here we use the Lie bracket in T(A, 1); see Definition 11.1.5.

Proof Let a, b E [XPh, 1MA], d E [XP,, I,MA] be elements which realize a, b, and d respectively. Then we have for the generators 62, h.h in Section 11.5 the formulas

µ([1,1](a#b)1=h h) = rh(a, b) (1)

µ(d42) = y(d). (2)

Here [1, 1]: IMA A MA - >.MA is the Whitehead square; compare the notation in (11.5.10) below. If h is odd letSh,h E7r4XPh n Ph = i/h *7L/h be the canonical generator. Now the James-Hopf invariant y3 satisfies the following distributivity laws:

y3([1,1](a#b)Ch,h)

Y- T221+7- TI21-Y' TI 12-Y.T212)a#41,h

= b#b#a(Y'T22I)Sh,htt + a#b#a(Y-T121 )6h.hCC

-a#a#b(1.T212)Sh.h -b#a#b(1T212)Sh.h =(h(h-1)/2)(b®b®a+a®b®a-a®a®b-b(9 a®b). 342 11 HOMOTOPY GROUPS IN DIMENSION 4

Here we have in (&3A the equation a®b®b=b®b®a+[[a,b],b]. Hence we have proved the first formula in Theorem 11.1.16. Moreover we get

Y3(d42) = (d#y2(d) + This yields the second equation in Theorem 11.1.16 since 2d=O. The equations above for y3 are obtained by the results in Section 11.5 below and the general distributivity laws in Section A.10.

We observe that 2y3 = 0. As an application we obtain the following result which we already mentioned in Theorem 11.1.9.

(11.1.17) TheoremFor any abelian group A the extension for ir4 M(A, 2) in (11.1.13)(3) is split. This implies that L(A,1)3 is a direct summand of 7r4M(A, 2).

Proof For any abelian group A we have the commutative diagram L3(A, 1)

®3A L3(A,1) [[1,11,1] where 3 denotes multiplication by 3. In fact, for x, y, E A we get [[1,11,1]i[[x,yl, z] = [[1,11,1](xyz +yxz - zxy -zyx) = [[x,y],z]+[[y,x],zl - [[z,xl,yl- [[z,yl,x] = 3[[x, y], z] Here we use [y, x] = [x, y] and the Jacobi identity. The diagram shows that the extension in the bottom row of (11.1.14) yields an element

( ®3A) E Ext(®2A/L3(A,1), L3(A,1)) of order 3, that is 3( (& 2A} = 0. On the other hand, the pull-back diagram (11.1.14) shows that the extension in the top row satisfies

{7r4M(A,2)} = E Ext(fT(A), L3(A,1)).

Here y3 by Theorem 11.1.16 satisfies 253 = 0. Hence we get (ir4M(A, 2)) = 0 and therefore the extension for ir4M(A, 2) is split. 11 HOMOTOPY GROUPS IN DIMENSION 4 343

(11.1.18) Definition We define for any abelian group A the retraction r: 7r4M(A,2) - L3(A,1) of the inclusion A by the formula

r(x) = [[1,11, lly3(x) - 2y3(X). (1) Here we use the James-Hopf invariant y3 in (11.1.14). Since 2y3 = 0 we see that 2y3(x) E L3(A, 1) C &A. Moreover for y E L3(A, 1) we get rO(y) = [[1,11,11y3A(y) - 2y3A(y) _ [[1,11,1]i(y) - 2y =3y-2y=y so that r is indeed a retraction for A. Here we need the commutative diagram in the proof of Theorem 11.1.17. Using the retraction above we obtain the isomorphism (µ,r): a4M(A,2) a FT(A) ®L3(A,1). (2) In case A is a direct sum of cyclic groups this isomorphism, however, is not compatible with the direct sum decomposition given by the Hilton-Milnor formula.

Next we compute the group is M(A, 2). For this we choose for each abelian groups A a map (16.1.19) 4: M(A, 2) -> M(A (9 71/2, 2) which induces the quotient map q: A --> A ® 71/2 in homology. If A is a direct sum of cyclic groups we choose 4 as a suspension. The map 4 induces the commutative diagram

(11.1.20)

IF(A) ®Z/2 >--°+ ir4M(A,2) µ - fT(A)

a 1q. 14. pull 1q. F(A ®1/2) ®Z/2 >-- ir4M(A (9 Z/2,2) -. I'T(A ®1/2)

The rows of this diagram are the short exact sequences given (11.1.13X4). Since q* on the left-hand side is an isomorphism this diagram is a pull-back diagram. For a basis B of the Z/2 vector space A ® Z/2 we get M(A(9 71/2,2)_ 'I XP2. B 344 11 HOMOTOPY GROUPS IN DIMENSION 4

Since7r41P, = Z/4 andir4EP2 A P2 = 1/4,seeSection11.5,the Hilton-Milnor formula in Remark 11.5.3 below shows that 4( V 1 P2) = U is a free Z/4 module generated by the disjoint union B u ((b,b'),b

(11.1.21) TheoremFor any abelian group A we obtain the abelian group 7r4 M(A, 2) by a pull-back diagram:

ir4M(A,2) U pull I 19 L3(A,1) ®fT(A)pFT(A)qI T(A ® 71/2) Here P2 is the projection and q: A -A ® 1/2 is the quotient map. Moreover U is a free 1/4-module for which there is an isomorphism 0: U ® 71/2 = I'T(A ® Z/2). This isomorphism defines q' = 0,:U- U® 1 /2 = rT(A ®1 /2).

11.2 On ir3(A, M(B, 2))

We now consider the homotopy group ir3(A,M(B,2)) of a Moore space M(B, 2) with coefficients in A where A, B E Ab. We have the universal coefficient sequence which is compatible with the suspension operator:

(11.2.1)

Ext(A,ir4M(B,2)) vr3(A,M(B,2)) -Hom(A,IB) 1: o Ext(A, Ir6M(B,4)) >-* Tr5(A, M(B,4)) -* Hom(A, B (9 71/2)

If B = 71/k is a cyclic group we derive from Proposition 11.1.12 that .; is an isomorphism. Hence in this case the diagram is a pull-back of abelian groups. In (8.2.12) we computed the bottom row of (11.2.1), hence (11.2.1) yields the group ir3(A, M(l /k, 2)) as a functor in M(A, 3) E M3. More generally we can compute the group ir3(A, M(B, 2)) as follows.

(11.2.2) DefinitionFor an abelian group B we have the surjective operator 11 HOMOTOPY GROUPS IN DIMENSION 4 345

µ: 1r3(FB, M(B, 2)) --* Hom(FB, I'B) as in the top row of (11.2.1) where we set A =1'B. A generalized Hopf map

rlB : M(IB, 3) --> M(B, 2)

is a map which satisfies µ(r18) = lre where 1rB is the identity of IB.If B is free abelian thenOB is well defined up to homotopy; in fact r)d = 02 is the Hopf map for B = Z.

A generalized Hopf map71B induces the commutative diagram (11.2.3)

Ext(A,r(B)®71/2) ° rr3(A,M(rB,3))

IA. push f(77B) . II Ext(A,ir4M(B,2)) - 7r3(A,M(B,2))-* Hom(A, FB) which is a push-out of abelian groups. Here 0* is induced by A in (11.1.8). We know the group (11.2.4) ir3(A, M(FB,3)) = G(A, FB) by use of the category G in Definition 11.6.6. Hence the push-out diagram (11.2.3) determines the extension in the bottom row completely. For example, if B is cyclic then FB has no direct summand Z/2 and hence the toprow of (11.2.3) is split and therefore also the bottom row of (11.2.3) is split. We have the composite

Ext(B,TZA) Ext(B, 7r4M(A,2))) ir3(B, M(A,2)) where F2A = F(A) ® 7 1 / 2 E D As in (11.1.13) we introduce two new functors

(11.2.5) 7r3',ir3: (M3)°P X MZ -> Ab 7r3(B, M(A, 2)) = rr3(B, M(A, 2))/00* Ext(B, F(A) (& 1/2) 7r3(B, M(A,2)) = rr3(B, M(A,2))/A0*Ext(B, L3(A,1)).

Here AA* for 7r3 need not be injective while 00 * for 7r3 is always injective since L3(A,1) is a direct summand of vr4 M(A, 2). The operator k= Q in Addendum 11.4.5 below, see also Section 6.6, yields the following natural diagrams in Ab in which all rows are short exact.

Ext(B, L3(A,1))Nrr3(B, M(A,2)) - FT,(B, A). (1) 346 11 HOMOTOPY GROUPS IN DIMENSION 4

This sequence is split for all A, B in Ab (unnaturally) as follows from the push-out diagram (11.2.3) and the fact that (11.1.13X3) is split for all A.

Ext(B,ir4M(A,2))°7r3(B,M(A,2)) Hom(B,FA) IA lExl(B,µ) (2) Ext(B, rT(A)) N TT#(B, A) -- Hom(B, rA)

Ext(B, F(A) (& Z/2)/K° 7r3(B, M(A, 2)) -A* rT,(B, A) (3)

Here K is the image of the boundary homomorphism

a: Hom(B, rT(A)) - Ext(B, r(A) ®1/2) (4) associated by the extension (11.1.13X4) for ir4M(A,2). The exact sequence (3) is a consequence of (2). As in (11.1.13X5) we have the binatural pull-back diagram

ir3(B,M(A,2)) -q 1r3(B,M(A,2)) IA (5) lq ir3(B,M(A,2)) rT,(B,A) where q denotes the quotient map. This shows that the induced maps on ir3(B, M(A, 2)) are completely determined by the functors ir3 and 7r3in (11.2.5). Similarly to (11.1.13) we have the following natural interpretation of the functor 7r3 above. Again let C M(A, 2) be the universal cover of the loop space of M(A, 2). Then we have the Hurewicz map with coefficients

ir3(B, M(A,2)) = ir2(B, f M(A,2))

= ir,(B,1 M(A,2))h >H2(B,flM(A,2)). (6)

Here the right-hand side is the pseudo-homology with coefficients in B. We obtain the isomorphism

hq-' : 7r3(B, M(A, 2)) = H2(B, f2M(A, 2)) (7) which is natural in M(A,2) and B. A proof of (7) is deduced from the generalized r-sequence with coefficients in the same way as in (11.1.13)(7). 11 HOMOTOPY GROUPS IN DIMENSION 4 347 11.3 On F4X and r3(B, X)

We here study the groups r4X and r3(B, X) of a space X but we do not yet determine the functorial properties of these groups. Recall that J.H.C. Whitehead described the group r3X by use of the functor r, that is (11.3.1) r3Xr(7r2X).

Hence 173X depends only on the second homotopy group of X. We know that 174X (as an abelian group) depends only on the quadratic function

(11.3.2) 7IX-7I2:7r2X-7r3X

induced by the Hopf map172 e ir3S2. In fact,71x determines the 3-type K(77x, 2) of the universal covering X of X, see Proposition 7.1.3, and hence 71x determines 174X =r4X = r4 K(71x, 2) as an abelian group. We now show how to compute 174X in terms of the function 71x.

(1133) DefinitionRecall that rAb is the category of quadratic functions; see (7.1.1). Objects are quadratic functions 71: A -> B between abelian groups which we can identity with homomorphisms 77r(A) --> B. We now define the functor F22: FAb - Ab (1) which generalizes the functor r2 in Definition 11.1.6. The functor r2 carries the object 77 to the abelian group 172(7,) given by the push-out diagram

B®Z/2®B®A 9 0 r;(11) 710 ®z/2o770®A1 push o. (2) F(A) ® 71/2 ®r(A) ®A - 172(A)

Here the bottom row is given by the quotient map in the definition of r2(A). A map p = ((po, pl): 71-q' in lAb with 77'Vo = (p,71 induces cp* : r2 (77) - 172(q') by gyp, ®7L/2 ®V, ®(po where (po: A -A' and gyp,: B -* B'. We can describe 172(71) also by the quotient r2(71)=(B®Z/2®B®A)/M(77) (3) where M(71) is the subgroup generated by the following elements with x,y,zeA:

(71x) ®x (4) [x, y],, 0 1 + (71x) ®y + [y, x],, ®x. (5) 348 11 HOMOTOPY GROUPS IN DIMENSION 4

Here we set [x, y],, = q(x + y) - q(y) E B and 1 E Z/2 is the generator. Clearly the kernel of q in the top row of (2) coincides with the subgroup M(71); this follows by the corresponding relations (1) and (2) in (11.1.6). If B = I'A and if q = yA is the universal quadratic function then rl ° = 1 is the identity of FA and hence T; (yA) = F2 (A) in this case by (2). Since 71x in (11.3.2) is natural in X we obtain by (1) the functor Top*/=-'Ab, XHF (fix) (6) which carries the homotopy category of pointed spaces to abelian groups. (113.4) TheoremLet X be a pointed space. Then one has the short exact sequence I; N I4X-. FT(7r2X) which is natural in X. Here we have rz (W) = (7r3(X) ® Z/2 ® 7r3(X) (& 7r2(X))/M(71x) and A carries x ® 1 to 0(x (& 1) = x74 and x ®y to the Whitehead product A(x (9y) = [x, y] where x E 7r3X, y E 7r2X. We point out that for ig = 77x the element [x, y], = [x, y] E 7r3(X) is the Whitehead product of x, y E 7r2(X). Therefore the relations (4), (5) in Definition 11.3.3 which generate Mix) correspond to the well-known formulas (11.1.8)(1X2). This shows that the homomorphism 0 in (11.1.4) is well defined and natural. Injectivity of A in Theorem 11.3.4 shows that the group F 2267x) is the subgroup (11.3.5) F (71x)=7737r3X+[7r3X,7r2X]Cf4X where + is the sum of subgroups (not the direct sum). We derive from Theorem 11.3.4 the next result on the operator Q in Theorem 6.6.6. For this we simply set X = K(A, 2). (113.5) CorollaryWe have H5 K(A, 2) = I'4 K(A, 2) = FT(A) and Q: 7r4M(A,2) -' H5K(A,2) = FT(A) is surjective and coincides with A. Clearly for X = M(A, 2) the exact sequence in Theorem 11.3.4 coincides with Theorem 11.1.9. Therefore the naturality of the sequence in Theorem 11.3.4 yields for a simply connected space X the following commutative diagram.

F2(71x) >. F4X FT(HZX)

(11.3.7) n.1push Ja. II F (A)Hir4M(A,2)-'` FT(A) 11 HOMOTOPY GROUPS IN DIMENSION 4 349

Here we have A = Hz X and a : M(A, 2) --> X is a map which induces the identity 1 = H2(a) on A. Moreover for r1=71x the induced map 71*in (11.3.7) coincides with q,, in Definition 11.3.3(2). Since diagram (11.3.7) is a push-out of abelian groups we can compute the abelian group r4 X by use of 7r4 M(A, 2). We determined the extension in the bottom row of (11.3.7) for any abelian group A in Theorem 11.1.21. Therefore the push-out diagram (11.3.7) shows how to compute the abelian group I'4X only in terms of 71x. In a similar way we can compute the group F3(B, X) for a simply connected space X since we have the commutative diagram

Ext(B,r4X) H f',(B,X) -.Hom(B,F3X) (11.3.8) TExt(B, a,) T-. II Ext(B,7r4M(A,2))- 7r3(B,M(A,2))-* Hom(B,FA) induced by a above. This is a push-out diagram of abelian groups and we know the extension in the bottom row by (11.2.3). This yields also the extension in the top row since we know how to compute a* in (11.3.7). In the rest of this section we prove Theorem 11.3.4.

Proof of Theorem 11.3.4Let H; = H; X and a; = 7r; X and let a be given as in (11.3.7). We consider the homomorphism

M:Tr3®71/2®7r3®7T,->r4X which satisfies M(x (9 1) =x714 and M(y ®z) = [y, z] for x, y E 7r3, Z E iT-,. Only the 4-skeleton of X is involved in the definition of r4 X. Therefore we may assume that X is a CW-complex with X 1 = * and dim X = 4. Let C* = C,, X be the cellular chain complex with cycles Z; = ker(d: C, - C,-,) and boundaries B; = dC;+,. Lett: B, -, C3 be a splitting of d so that C3 = tB2 ® Z3. Then there is a map

(1) g: A =M(C4,3) VM(tB,,2) -+B=M(Z3,3) VM(C2,2) such that the mapping cone Cg of g is homotopy equivalent to X, see 1.7.5 in Baues [CH]. The map H2g coincides with the inclusion tB, - C; given by d. Moreover the map Hag coincides with the map d: C4 - Z3 given by d. Hence g is determined up to homotopy by the triple (C, t, g) where g: C4 - r(C,) is a homomorphism given by the coordinate M(C4, 3) -i M(C2, 2) of g. We consider the commutative diagram

d a 7r5(Cg, B) 7r4 B r a 7r4 Cg/ a7r4(Cg, B) Tr3B (2) E8fi E1 7r4(AvB)2 7r3(A V B)2 350 11 HOMOTOPY GROUPS IN DIMENSION 4 the top row of which is the exact sequence of the pair (Cg, B). The operator EE = (irg,1),, a-' is the functional suspension in (11.1.10).

(3) LemmaBoth homomorphisms Eg in (2) are surjective

Proof The result is clear by Corollary A.6.3 for n = 3 since A is 1-connected (a = 2). For n = 4 we consider the exact sequence in Theorem A.6.9 which is the row in the following commutative diagram ker(g,1) - jir4Cg

E8 HH P8 r Eg r 7r4(A V B)2 - 7r5(Cg, B) -s 1r3(A AA') 'r3(A V B)2 -+ a4(Cg, B)

al (g, 1). (B) Here we have A = )A'. We show below that Pg is injective, therefore Hg = 0 and thus Eg on the left-hand side is surjective. By definition of A and of Pg = [i1, i - i2g]* the following diagram commutes ir3(A AA') -- ir3(A V B)2

11 II tB2 ®zB, - 8 - C4 ®r (tB2) 9 tB2 ®C2 with Pg = (0, [1, 1], - 1 ® d). Since d is injective also 10 d is injective and thus Pg is injective.

(4) CorollaryWe have the short exact sequence Vg- Tr4Cg-.Ug where Vg = iir4 B = ir4 B/(g,1) 7r4(A V B)2 and Ug = jir4Cg = ker(g,1) * /Pgir3(A A A').

(5) Lemma For 1r; = rrCg we have Vg = ia4B = ('13)`ir3 + [ir3, Tr2]. This is the image of M above.

Proof of (5)Compare diagram (6) below. We know that ir3B --I- 7T3C9 and ire B --> 1r2Cg are surjective. Therefore Image M C Vg. On the other hand, we have by the Hilton-Milnor theorem the surjection M=MB: ir3B®7L29 7r3B®a2B-.lr4B. This shows Vg c Image M c 1r4Cg. 11 HOMOTOPY GROUPS IN DIMENSION 4 351 We consider the diagram

Ir3Cg ® 711 ® Ir3Cg ®Tr,Cg M 7r4Cg

(6) T'a

7r3B ®71, ®-rr3B ®1r,B -B»7r4 B

Here i = i3 ® l2 +r3 ® i2is surjective where r,,: rrnCg is induced by BcCg.

(7) Lemma ker M = i ker MB.

This is the crucial fact which we use for the proof of Theorem11.3.4.

Proof of (7)Consider the following commutative diagram with the notation below. M. (a) -77r3B®71291T3B®7T2B yr4B 1L (g.1). G ------+ 7T4(AVB)21 n n 7T3(AVB)®Z2®7r3(AVB)07r2(AvB)* or4(AVB) MA,8 Here (g,1)*=(g,1)* ®712®(g,1)* ®(g,1)*. Since ker i4= (g,1)*ir4(A vB)2 the lemma follows from

(b) iL=Oif 7T4(A V B)2. In fact, we have by (6) the inclusion i ker R. c ker M. The inclusion ker M c i ker MB follows from (b) bya diagram chase in (6) and (a). We obtain G in (a) and (b) by G=(zr3(AvB)2(& 712)ED (Tr3(AVB)2(& ir,(AvB))®(ir3B®ir2A). Now iL= 0 follows from the following exact sequences: 1r3(A V B)2 7r3(B) --a ir3Cg, (g, I). ire (A) B 7T2( B) ---- a2Cg . 0

Next we show:

(8) Lemma The kernel of M is generated by the elements in Definition 11.3.3(4), (5). 352 11 HOMOTOPY GROUPS IN DIMENSION 4

Proof of (8)For MB we have the following description as a direct sum of homomorphisms:

Z3®71, I Z3 ® 712

MB = Z3®C2 1 z3®C2 7r3B0®Z2®7r3B0®7r2 BO 7r4 B0 MB0

Here B0 = M(C2, 2) and MB = 1 ® 1 ®MBo. This shows that

(9) kerMB = kerMB..

Now (7) and (9) show that it is enough to prove (8) for X = B0. This can be done by use of the Hilton-Milnor theorem. In fact, we choose a basis Z in C2 with

(10) X=Bo=M(C2,2)= V S2. z Moreover, we choose an ordering of the basis Z. Then 7r4 X is the sum of the following cyclic groups with generators as indicated and 71= 712 E 7r3S2,

712 = 7r4(S2) E) a71(E71) with a E Z, b E Z (11) 12 = 7r3(S3) E) [a, b](E71)with a

On the other hand, 7r3X = F(C2) is the free abelian group generated by y(a) and [a, b] with a < b. Clearly, 7r2 X = C2 is the free abelian group generated by the elements in Z. We have to compute the kernel of

M: I'(C2)®Z2®F(C2)®C2-'>7r4X

y(a) ®1 --i a71(En) [a, b] ®1 -+ [a, ME77 (a < b) (12) y(a) ®b H [a71,b] [a,b] ®c H [[a,b],c] (a

The Barcus-Barratt formula yields

(13) [a71, b] = [a, b](E71) + [a, [b, all 11 HOMOTOPY GROUPS IN DIMENSION 4 353

Since [r), i2] = 0 in 1r4(S2) we have [arl, a] = 0. Now (11), (12), and (13) imply that ker M is actually generated by the elements in Definition 11.3.3(4), (5). This is seen by expressing the values of M in (12) in terms of the generators in (11). For this we use (13) and the Jacobi identity for Whitehead products. This completes the proof of (8).

For the proof of Theorem 11.3.4 we show that we have the commutative diagram

z(,qX> r2 r4Cg - rT(H2)

(14) Vg H'r4 Cg Ug 1h 1h ker b4 = ker b4 in which each row and each column is a short exactsequence; compare Theorem 11.3.4 and (4). By (4) we have

(15) Ug = ker(g,1)*/Pgir3(A A A') where we use the following homomorphisms

P (g. U. Ir3(A AA')` 1r3(A V B)z

II II tB2 ® tB2 ->c4®r(tB2)6) tBZ®c2-,Z3®rc2.

Here we have

Pg, =(0,[1,1], - 1 ®d3)

(g,1)* _(d4, g) ®(rd3, [d3,1]). This shows by definition of rT(H2) in (11.1.1) that the kernel of h: Ug ker b4 is rT(H2). Recall that b4 makes the diagram

C4 -8-a r(c2) U iP Hob r(H2) commute where ker p = im (rd3, [d3,1]) and where g is defined by g in (1). 354 11 HOMOTOPY GROUPS IN DIMENSION 4 11.3A Appendix: Nilization of F4X

We consider the functor

(11.3A.1) nil: Gr - Gr which carries a group G into its (second) nilization nil(G) = G/I'3G where r3G is the subgroup of G generated by triple commutators in G. Clearly the abelianization satisfies A = Gab = (nil G)ab. For a free group G we have the natural short exact sequence

(11.3A.2) A2(A)rnil(G) ab»A where ab is the abelianization and where w is the commutator map with w((x) A [y)) = -x - y + x + y for x, y e nil(G) and {x) = ab(x). Clearly the exact sequence (11.3A.2) fits into the commutative diagram of short exact rows of groups -a'-- [G,G] H G A

(11.3A.3 ) Inil Inil II A2(A) Nnil(G)-. A where nil is the quotient map and where [G,G] is the commutator subgroup of G.

Now let G = GX be the simplicial loop group of Kan; see (11.4.7). Then A = AX is the abelianization of GX as in Theorem 1.4.8 and in this case (11.3A.3) is a diagram of simplicial groups which induces the natural operator

(11.3A.4) _,[G,G] rr-,AZ(A) = r,°il(X) which we call the nilization of r (X). Here we assume that X is simply connected. For n < 4 we compute the nilization r,"(X) as follows. We have the exact sequence

(11.3A.5) F(H7X)-4 7r3X--,H3X->O which is part of Whitehead's F-sequence. Here q ° is induced by r, = -1x in (11.3.2). We therefore can identify the third homology H3X with the cokernel of i7x . The definition of r2 267x) in Definition 11.3.3 shows that the sequence (11.3A.5) induces the exact sequence

(11.3A.6) r2(H2X)-n`->FZ(n,,)"H3X®Z/2®H3X®H2X- 0 11 HOMOTOPY GROUPS IN DIMENSION 4 355 where It * is given by h ® 1/2 ® h ® H2 X. Using h* we obtain the following push-out diagram which defines r4''(X)

r4(X) rT(H2X) (11.3A.7) ,h, push jh II H3X®Z/2®H3X(&H2X -I;"(X)-.rT(H2X)

The push-out (11.3.7) and (11.3A.6) show that the bottom row splits (unnaturally). Moreover we have by (11.3A.6) the natural exact sequence

(11.3A.8) >I;''(X)->0.

(11.3A.9) TheoremLet X be simply connected. Then the nilization

nil * : r (x) -> I'"" (X) is an isomorphism for n < 3 and for n = 4 one has a natural isomorphism 0 for which the diagram r;i'(X) nil r4(x)/ e 1r4i'(x) commutes. In particular nil * on r4(X) is surjective and the kernel is determined by (11.3A.8).

We think of diagram (11.3A.7) as the unstable analogue of the diagram in Theorem 5.3.7(a), so that r4''(X) is the unstable analogue of the group H3(X,71/2). Indeed the suspension o, yields the commutative diagram

(11.3A.10) H3X®Z/2®H3X®HZX) rn"X rT(H2X) to l(l,0) 1- H3X®71/2' ' -+H3(X,l/2) -.H2(Xy)*71/2

Here the bottom row is the universal coefficient sequence. All vertical arrows are surjective. The unstable analogue of the diagram in Theorem 5.3.8(a) is 356 11 HOMOTOPY GROUPS IN DIMENSION 4 the following commutative diagram for the secondary boundary b5 of X

ker Sq' N H5 X

(11.3A.11) F(H,X) , r4 X r4 aX ker (rl* n

This diagram defines the natural maps Sq' and 0' for simply connected spaces X. The stabilization of diagram (11.3A.11) is the diagram in Theorem 5.3.8(a). Therefore Sq' is a desuspension of the Steenrod square Sq2 and 0' is a desuspension of the secondary Adem operation ¢* , that is rSq' = Sq2 and oo' = 0*.

11.4 On H3(B, K(A, 2)) and difference homomorphisms

We determine the pseudo-homology H3(B,K(A,2)) and we describe the connection of this group with Tr3(B, M(A, 2)). This also leads to the computation of difference homomorphisms for induced maps on ir4M(A,2) and 'r3(B, M(A, 2)). The quadratic functor r: Ab - Ab yields the bifunctor

(11.4.1) rT#: Ab°P X Ab -p Ab which carries a pair of abelian groups (B, A) to the set of homotopy classes of chain maps rT#(B, A) = [dB, r*dA]. (1) Here dA: AI 'A0 is a short free resolution of A which is a chain complex concentrated in degree 0 and 1. Moreover r* dA is the chain complex in (11.1.1); see also Definition 6.2.11. We have FT,,(7L, A) = r(A). In fact, since Hor* dA = F(A) and H, r,, dA = FT(A) we get the binatural short exact sequence

Ext(B,rT(A))>-'->FT,(B,A) - Hom(B,r(A)) (2) which is split (unnaturally). Here µ carries a chain map in (1) to its induced map in homology. We now consider the pseudo-homology H3(B, K(A, 2)) of an Eilenberg-Mac Lane space K(A, 2) given by the group of homotopy classes of chain maps, H3(B, K(A,2)) = [C* M(B,3),C* K(A,2)), where C, is the singular chain complex. 11 HOMOTOPY GROUPS IN DIMENSION 4 357

(11.4.2) TheoremThere is a binatural isomorphism H3(B, K(A,2)) = rT#(B, A) for which the following diagram of short exact sequences commutes.

Ext(B,H5K(A,2)) * H3(B,K(A,2)) - Hom(B,H4K(A,2))

II II µ II -» Ext(B,rT(A)) A H I'TT(B,A) Hom(B,I'(A))

Here we use the isomorphisms r(A) = H4 K(A, 2) and HS K(A, 2) = 17(A ). In the proof of Theorem 11.4.2 we use the surjection Q in Corollary 6.6.8 and we obtain a new interpretation of Q by use of twisted maps between mapping cones. To this end we describe the following concept of twisted maps which generalize the principal maps in Definition 6.12.11.

(11.4.3) DefinitionLet f: X --> Y and g: U - V be maps in Top*. A twisted map F=C(u,c,H,G): Cf->C5 (1) between the mapping cones of f and g is obtained as follows. Consider a diagram X -UvV fl I(g,l) (2) Y--> V together with homotopies H: rf = (g, 1)u and (0, 1)u = 0. Then there is a pair map G: (CX,X)-*(CUVV,UvV) extending u. Here CU is the cone on U. Using the inclusion ig: V C Cg and the projection (7rg,1): CU v V --i Cg, see (11.1.10), for the mapping cone Cg we get F in (1) by the formulas: F(y)=igr(y) foryEY F(t, x) = igH(2t, x) for 0< t < 1/2, x E X F(t, x) _ (rrg,1)G(2t - 1, x) for 1/2 < t < 1, x c= X.

Let TWIST(f,g)c[Cf,Cg1 (3) 358 11 HOMOTOPY GROUPS IN DIMENSION 4 be the subset of all homotopy classes represented by twisted maps. Moreover let

Twist(f, g) c [ X, U v V12 X[ Y, V ] (4) be the set of all pairs of homotopy classes (u), (v) as in (2) with (g, 1) * (u) = f *{v} and (0, 1) * {u} = 0. We say that F in (1) is associated to the pair ({u}, (v}) E Twist(f, g). The properties of twisted maps are studied in Baues [AH], Chapter V. The Moore spaces M(A, 2), M(B, 3) are mapping cones of dA: M(A 1, 2) -- M(Ao, 2) and dB: M(B1, 3) -> MOO, 3) respectively. Now it is easy to see that

(11.4.4) Twist(dB, dA) = Chain(dB, r* dA) where the right-hand side is the set of chain maps dB - I * dA. In addition to Theorem 11.4.2 we show the

(11.4.5) AddendumThere is a commutative diagram

[ M(B, 3), M(A, 2)]-H4(B, K(A, 2))

II II A TWIST(dB,dA)3 2 [dA,f*dB ]=FT#(A,B) where k carries a twisted map F associated to (x, y) to the homotopy class of the chain map (x, y) given by (11.4.4).

Proof of Addendum 11.4.5 and Theorem 11.4.2The arguments in (V.§7) of Baues [AH] show that all maps M(B, 3) - M(A, 2) are homotopic to twisted maps. Moreover (V.7.17) in Baues [A]H] shows thatk is well defined. The push-out diagram in Corollary 6.6.8 yields the kernel of Q. Since A fits into a push-out diagram of the same kind we see that kernel(A) = kernel(Q). We now define the isomorphism in Theorem 11.4.2 by AQ-'.

We need the following purely algebraic notation.

(11.4.6) DefinitionLet A, ir, R be abelian groups and let CE Ext(A, ir) be represented by 6, E Hom(A1, ir) where dA: A, -->A0 is a short free resolution of A. Then we obtain the composite

fi®R t*: A*RcA,®R - rr®R (1) 11 HOMOTOPY GROUPS IN DIMENSION 4 359 which depends only on ; see also Definition 8.3.11. Moreover we obtain the commutative diagram Ext(B, A * R)

[dB,dA ®R] Ext(B, a® R)

Here # in the bottom row is defined as follows. For a chain map F: dB --> dA ® R the element 4{F} is represented by the composite B, F^AR The inclusion A in (2) is the usual one which carries (b) represented by b E bHom(B1, A A * R) to the chain map which is 0 in degree 0 and which. is B, - * R cA, ® R in degree 1. Hence diagram 2 commutes. If A and B are finitely generated we have a natural retraction r: [dB, dA ®R] -4 Ext(B, A * R) (3) of A in (4); see Lemma 6.12.13. In this case f,,in the bottom row of (2) is simply Ext(B, C,)r. Using Addendum (11.4.5) we are able to prove the unstable analogue of the deviation formula in Proposition 8.3.12. For this consider the elements a,a+D&E[M(.A,2),X] withf E Ext(A, IT3X). Here a + A6isdefined by A: Ext(A, rr3X) - ir2(A, X). We obtain the difference homomorphisms (a + A C) a * r4(a+06)-r4a: r3(B,a+ze)-r3(B,a): a3(B,M(A,2))- r3(B,X).

Here r4a = a*is the same as in (11.3.7). Let 7T;= zr; X and let a= ir,(a) E Hom(A, ir2X) be induced by a.

(11.4.7) TheoremThe difference homomorphisms above yield the following commutative diagrams (11.4.8) and (11.4.9). o.H Tr4M(A,2) µ FT(A) A *(71/2 ®A) lt. I (11.4.8) (a+ A0* -a. 73 ® (Z/2 ®A)

r4(x) ° -rZ(17X) 7r3®(/2®ir2) 360 11 HOMOTOPY GROUPS IN DIMENSION 4

Here we use H) in (11.1.4) and the quotient map q in Definition 11.3.3(3); moreover we set a * = 7r3 0 (Z/2 ® a).

(11.4.9) (Q,h). 7r3(B, M(A, 2)) FTO(B, A) [dB, dA (9 U/20 A)]

(a+A f). -a. Ext(B, 7r3 ® ( 7 1 / 2 ®A))

I Ext(B,A . ) A f3(B, X) Ext(B, F4X)

Here (o, h),, is induced by the chain map (Q, h): F* dA --> d® ® (71/2 ®A), see (11.1.4). The surjective homomorphism A is defined in Addendum 11.4.5. If A and B are finitely generated we can use the retraction r in Definition 11.4.6(3) for

Proof of Theorem 11.4.7For the mapping cone Cd = M(A, 2), d = dA, we know that each element x E 7r4(Cd) is functional in the sense that j(x) _ (7rd, I) * a- `Q) = see (11.1.10). Hence 11.12.3 in Baues [AH] shows

(1) where ao = a I M(A0, 2) and where 6,: -I X represents . Moreover E is the partial suspension for which the following diagram commutes

E 7T3(X, V XO)2 -> 7r4(Y'Xt V X0)2

(2)

FA, ®A, ®Ao (D A, ® 71/2 ®A, ®A0

Hence (1) shows that the first diagram of Theorem 11.4.7 commutes.

Next we know by Addendum 11.4.5 that F E 7r3(B, M(A, 2)) is a twisted map associated to (u, v) E Twist(dB, dA) = Chain(d8, F dA). Hence we have by 11.12.7 and V.3.12 (3) in Baues [AH] the formula

(3)

Now one can check as in (2) that Eu is given by (Q,T)A(F) with A(F)= ((u, v)). This yields the commutativity of the second diagram in Theorem 11.4.7. 11 HOMOTOPY GROUPS IN DIMENSION 4 361 11.5 Elementary homotopy groups in dimension 4

Let PCyc° be the full subcategory of Ab consisting of elementary cyclic groups 71 and 71/p` where p' is a prime power. We also write 71= 7/0. The elementary Moore spaces are (d >_ 2)

(11.5.1) M(Z/n, d) _ 1d- `Pwith71/n E PCyc°. Here P = S' U nee is the pseudo-projective plane for n not equal to 0 and P0 = S'. The quotient space X A Y = X X Y/X V Y is the smash product for pointed CW-complexes X, Y. The elementary homotopy groups which we consider in this section are for 71/k,71/n,71/m E PCyc° the groups

(11.5.2)

7r4Y.P,,,ir4Y.P APm, ir3(71/k,IP AP,,,). The elementary homotopy groups arise in the following application of the Hilton-Milnor theorem.

(1153) Remark Let A and B be direct sums of cyclic groups

A = (B (71/a;) a;andB = ® (71/b,)13i i i with 71/a1,71/bi E PCyc°. Then we obtain the Moore spaces M(A,2) _ V IPa , M(B,3) _ V 12Pb, i S as one-point unions of elementary Moore spaces. The inclusions a;: IPa, C M(A, 2) yield the Whitehead product [ a;, ai ]: Y.Pa A Pa -> M(A, 2). Then the Hilton-Milnor theorem shows

(D ir,1Pa) ® (® ir4Y.Pa n Pa) ®L3(A,1). Tr4M(A,2) «i The isomorphism is given by 7r4(a;) on 1r4I Pa , by a4[ a;, ai ] on a,Pa A Pa , and by 0 in Theorem 11.1.9 on L3(A, 1). Similarly we get

ar3(B, M(A,2)) ® 7r3(B, Y-Pa)) ® 7r3(B, YPa A Pa))

a) Ext(B, L3(A,1)). This shows that the groups ir4 M(A, 2) and rr3(B, M(A, 2)) are completely determined by the elementary groups in (11.5.2). 362 11 HOMOTOPY GROUPS IN DIMENSION 4

(11.5.4) Elementary exact sequencesLet Z/n,7L/m E PCyc°. Then we have short exact sequences:

F(71/n) ® Z/2N7r41 P,_gFT(71/n) = 7L/n *1/2 (1)

µ 71/n*i/m (2)

Ext(B, -rr4EP,,) N 7r3(B, -go Hom(B, I'(7L/n)) (3)

Ext(B, ir4IP A P H .Tr3(B,!,P A -Hom(B,7L/n ®7L/m).(4) Here (1) is a special case of Theorem 11.1.9 and (2) is part of Whitehead's exact sequence F4 X -* 1r4 X -> H4X for X= I P, A Pm . Moreover (3) and (4) are for B E Ab the universal coefficient sequences. We solve the extension problems for the elementary exact sequences as follows.

(11.5.5) TheoremLet 71/n,71/m E PCyc°. Then (1) is non-split if and only if n = 2. Moreover (2) is non-split if and only if n = m = 2. Nert (3) is split for all B E Ab. For B = 71/k E PCyc° the sequence (4) is non-split if and only if k = 2 and (n, m) E {(2, 0), (0, 2), (2`, 2), (2, 2`), t > 1}.

(11.5.6) Remark Let A and B be direct sums of cyclic groups as in Remark 11.5.3. Then the extension in Theorem 11.1.9,

t2(A) ° 1r4M(A,2) - FT(A), coincides via Remark 11.5.3 with the direct sum of elementary exact sequences. Hence Theorem 11.5.5 also solves the extension problem for ir4 M(A, 2). Similarly Theorem 11.5.5 solves the extension problem for

Ext(B,Tr4M(A,2))4 ir3(B,M(A,2)) Hom(B,FA) since this sequence via Remark 11.5.3 is a direct sum of elementary exact sequences.

Proof of Theorem 11.5.5 We obtain the result on (1) by Proposition 11.1.12 and Theorem 8.2.5. Moreover for (2) we observe that ir4 X with X = I P A P. is in the stable range so that we can apply (5.3.5) where we use the Cartan formula for the computation of Sg2(i). Next the splitting of (3) is proved in (11.2.4). Finally using Remark 11.5.3 we solve the extension problem for (5.4X4) by the push-out diagram (11.2.3). Here we also can use Theorem 1.6.11. 0 11 HOMOTOPY GROUPS IN DIMENSION 4 363

Theorem 11.5.5 determines the elementary homotopy groups (11.5.2) as abelian groups completely.

(11.5.7) Remark The space P A Pm is also the mapping cone of n A 1: S' A Pm --> S' A Pm where n: S' - S 1is a map of degree n and where 1 is the identity of Pm with n, m > 0. The identity I,,, of IPm with m a prime power satisfies in[Y.Pm,1Pm] if m = 2 and (4, n) = 4 or if m not equal to 2 and (m, n) = m; see Corollary 1.4.10. In these cases we get the homotopy equivalence

P,, APm=IPm VI2Pm. For n = 2 = m there is no such decomposition since then P2 A P, is the mapping cone of

(2i,,!,i 2 + 213): S2 VS3 _* S2 V S3.

Here i, (t = 2,3) is the inclusion S` CS2 V S3 and n2 E= '7r3 S2 is the Hopf map. Compare IV.A.13 in Baues [CH]. For the elementary Moore spaces we use the inclusion and the pinch map

Sd `_i,, y d - (11.5.8) q-9^Sd+l and we use the inclusion

i=inm =Yi,, Aim: S3CYP APm. Let . E 'nr+ IS be the Hopf map with 1r?, = n,+ 1for n >_ 2. We write C = (7L/t)x if C is a cyclic group isomorphic to Z/t with generator x, t > 0. We know that

S2 173 = 7772, Tr.+ 1S'= (71/2)x1,forr >_ 3,

ir4S2 = (Z/2)172with112 =713712,

rr3Y-P =71/(n2,2n)ixlzand7r4E2P=71/(n,2)i713,

[y2Pk,S2] =Zl(k,2)q2qand[12p" S3]

[12p",I2p, = G(71/k, Z/n); see Theorem 1.6.7. Here we assume that n and k are powers of primes and (a, b) denotes the greatest common divisor of a, b E Z. These groups together with the groups in the next theorem yield a complete list of all elementary 364 11 HOMOTOPY GROUPS IN DIMENSION 4 homotopy groups in (11.5.2). For this we point out that the essential features of the elementary homotopy groups in (11.5.2) only arise when k, n, m are 0 or powers of 2. This follows from the short exact sequences (11.5.4) above.

(11.5.9) TheoremLet n, m, k be powers of 2. Then one has generators e, y, , r, for which the following equations hold. For the definition of these generators see (11.5.16) below. f 7/4)6 (26=e)forn=2 aa"= SI (Z/2)60 (7L/2)e for n4.

Here we write 6 = and e = e = i1722is given by the double Hopf map. f (7L/4)6 (2f=e) forn=m=2 (7L/2)e forn >_ 4 or m >_ 4.

Here 7L/r = 7L/n * 7L/m is the cyclic group of order r = (n, m). We write and e = em =L7j3 is obtained by the Hopf element. (7L/2) C ®(7L/2) y (--=O) fork=n=2 [X2Pk,ypl _ (7L/4)C®(Z/4)y (e=2C) forkz4,n=2 (7L/2) C ® (7L/g)y (D (7L/2)e for k2, n 4. Here 7L/g = Hom(7L/k, fl/n)) is the cyclic group of orderg = (2n, k) and we write y = y k. The elements f and a are given by C = rt =q and e = Lrl2q

[ 2Pk,Y,P APm]

(e=0) for (k, n, m) = (2,2,2) (--=2,q) for (k, n, m) _ (2, >_ 4, 2), (2, 2, >_ 4) (Z/4)6®(l/2)q (e=26)for (k, n, m) = (>_ 4, 2,2) (7L/e) f ® (i/d),q ® (7L/2)e otherwise. Here 7L/e = Ext(l/k,7L/n * Z/m) and 7L/d = Hom(7L/k,i/n (&7L/m) and we write 77 = rtk m. The elements f and e are given by ,m = mq and "73 q We need the following notation and facts; see Appendix A. Let A and B be finite dimensional connected CW-complexes with base point and let (11.5.10) T,,: A AB ->B A A be the interchange map. Moreover for maps f: !A --* IA', g: F,B -- EB' between suspensions let f#g be the composite f AB Y,A A B IA' A B T" IB AA' g- ,B' AA' I T21 ) Y.A' A B'. (1) 11 HOMOTOPY GROUPS IN DIMENSION 4 365 Hence f #g = If ' A g' if f = If ' and g = I.g'. Moreover for a E [ IA, X] and 16 E [IB, X] let

[a,f3IE[IAAB,X1 (2) be the Whitehead product. We have the interchange rule

[a, /3] _ -(YT21)*[ 9, a] (3) and for a E [A, Al b E [B, B'], v E [X, X'] we have naturality:

[(Ea)*a, (Ib)*f31 = (1a A b)*[ga, /3l v* [a, f31= [v* a,v*,6 1. (4) Moreover let A = AA: A -A AA be the reduced diagonal with A(x) =x Ax for x (=-A. Then one has

[a, P, + 1621 =[a, 621 +1 a,1611 +[1 a,161 ],1621I (A A AB). (5)

[a1+a,,f31=[a,,161 -[a2,[a1,16 1]1(AAAB)+[a,,161. (6)

We also write A A AB = T122 and AA A B = T112. Let A'"=AAA n ... n A be the n-fold smash product of A. For rp E [IB,IA] let

yn(tp) E [IB, IAA"] (7) be the nth James-Hopf invariant (defined with respect to the lexicographical ordering from the left). The James-Hopf invariant is natural in A and B. As an example one has Y2602) = 1 E 7r3S3 for the Hopf map rl2 and 21.12 = [1, 1]. We need the following formulas for the Whitehead square which is the Whitehead product given by the identity 1 of IA. If A is (r - 1)-connected and dim(A) < 4r - 3 we have

Y2[1,1] _ -IT21 + IT12 E [IA"2, IAA2] (9)

Y3[1,11=ET221+YT121-IT112-IT212E[IA n2,IA n3]. (10)

Here T,, is the identity of A A 2 and

T22,(a, A a2) = a2 A a, A a,, T121(al n ag) = a, A a2 A a,, and T212(al A a2) = a2 A a1 A a2 for a,, a2 EA. See Section A.10. 366 11 HOMOTOPY GROUPS IN DIMENSION 4

(11.5.11) TheoremThe James-Hopf invariants

Y3. ir4(Y.PR) - ir4(EPR AP APR) =71/n 11Pk,1PR] , [ZPk,Y.PR Y3 APR APR =Ext(71/k,71/n) are trivial for all n, k >_ 0.

Proof We may assume that n and k are powers of 2. Let X,, E [I P2, I PR ] be defined as in (11.5.16). Then we have the following commutative diagram with short exact columns

Y3 7r4S2 1T4S4 = 71

fiX,2).

ir4Y- P2 -IT4 Y' P. 7r4IP.AP.APR

I I I'T(71/2) =FT(71/n)

Since Y3( itis hence enough to prove y3a4P2=0' Now y3 is induced by the James map

n Y.P2 = JP2 - Y -> JPZ A P2 A P2 =11IP2 A P2 A P2 and we have the commutative diagram

ir3JP2 .L H3CJP(P2)-- H3JP2

Y.1 IY3

'7r3JP2 A P2 A P2 H3 JP2 A P2 A P2

Here we use as in Baues [CHI (111.6.6) the Hurewicz map for the universal covering JP2 of JP2:

h : ar3 JP2 = ir3JP2 -. H3JP2 = H3CJp(P2 ) which is surjective since fP2 is 1-connected. Hence it is sufficient to prove that the composite

H3CJp(P2)P,H3JP2-11 H3JP2AP2AP2 is trivial. This is an exercise. Here CJp(P2) is a completely algebraic chain complex; the map p corresponds to the projection JP2 --+ JP2 of the universal 11 HOMOTOPY GROUPS IN DIMENSION 4 367 covering. Next we consider y3 on We obtain the following commutative diagram.

Ext (7L k, rr41 73. P P A P,)

r II

;.3 `-

Here y3is trivial on Therefore there exists the factorization y3 which is natural in Z. and in Z,,. For n < k the inclusion X: 7L --> 7Lk yields the commutative diagram

Hom(7Lk, FZ) --Y-% Ext(7k, Z ) 3 1X 1 Y3 HOm(Z,,, Ext(i,,, Z,,)

Next consider the commutative diagram [1,1]* =o Ext(7Lk,1r4I,P A Ext(71k, ir4Y-

r a,tl. 2 I [y2Pk,Y,Pn [E

1 "--W---4 ! Hom(Lk,7L (& 11,11.

Here [1, 1] * is trivial on ir4MP n P,, by Lemma 11.5.15 so that the map W is defined. We observe that for k < n the map [1,1] * in the bottom row is surjective. Hence 7Y3 = 0 is a consequence of the fact that 73[1,1] by (11.5.10) (10) induces the trivial map on [2Pk,Y.P, A see Lemma 11.5.27.

We now consider the generators e, y, C, ,q in Theorem 11.5.9. First we recall that the identity I2 = X2 of IP2 satisfies

(11.5.12) [YEP2, Y.P2] = (71/4)X2with2X2 = '1129 We choose the generators 2 and f2,2 as in the following theorem.

(115.13) TheoremLet f2 be a generator of

-rr4Y.P2 = (71/4)f2 (1) 368 11 HOMOTOPY GROUPS IN DIMENSION 4

Then C2 induces the isomorphism of groups

z : [YP2,T-P21 = ir41P2. (2) Moreover the James-Hopf invariant

Y2: 1r4EP2 = IT4I P2 A P2 (3) is an isomorphism. Let z2 = Y2 62 -Then the following formulas are satisfied:

(4) 713 and262 = i7122 and262,2 ='713

(IT21)*62,2=62,2and (5) Proof The map in (2) is a homomorphism since the suspension I in z = 1062)*E isan isomorphism. Hence2is an isomorphism with z Xz = 62 Since q62 = 713 by (11.5.4) (1) we get

2f2=fz(2X2 _62* '712q=tg2g62 =1712713 =ii

2 2,2 = Y2(2i;2) = Y2 (1772 773) =1713 so that (4) is proved. We obtain (3) by the commutative diagram of elementary exact sequences

ir3(1P2) ®71/2 N 1r4I P2 - FT (71/2)

(11.5.14) ;E 17z®1 s lh 1Y= 7r3(IP2 AP2) ®71/2 >-> 7r4IP2 AP2 -µ 71/2*71/2 Moreover (5) is a consequence of the next lemma.

(11.5.15) Lemma For n >_ 0 the interchange map T21 induces the identity id=(IT21)*: -rr47. PnAPR-1r4I P,APR. Moreover the Whitehead square induces the trivial map 0= [1,11*: 1r4IP,, APn - 7r4IPn.

Proof We consider the following commutative diagram obtained by applying the suspension to Whitehead's exact sequence with n even

h 71/2n=F(H2P,,APn) 1T3PP AP. -' H3PP AP. =71/n

' h II 1° f 71/2=71/n®71/2 >-+ 2r4IP,, APn -' H4IP,APn 11 HOMOTOPY GROUPS IN DIMENSION 4 369 Now T21: P A P -- P,, A P induces the identity on H3 P A P. and on r(H2P Therefore there is a: 71/n -->71/2n such that

1r3(T21) = id + iah. This implies 1r4(IT21) = id + i(oa)h where, however, a-a = 0. Hence the first proposition is proved. We now prove 0 = [1, 1] * as follows. Since the genera- torc2 E 1r2S2satisfies[[a2, i2], c2] = 0 we seethat[[1,1],1] 0 = 0 for 13 E 1r4Y.P A P A P. Therefore we obtain for a E 7r41 P,,the equation a = 2a - [1,1]y2(a) where I. = X,^is the identity of Y,P,,. Here we use (11.5.11) and id = (IT21) * above. Now let n = 2 and a = 62. Then we get by Theorem 11.5.13(4) and (11.5.12).

C C (212) * 1tS2 = l7)2gS2 = 111283 = 262 This implies [1,1]y2 61 = 0 and hence 0 = [1,1]* for n = 2. Next let n = 21, t > 1, and let X: 71/n - Z/2 be the surjective homomorphism. We choose X: P --> P2 which induces X = Now we obtain the commutative diagram of elementary exact sequences

1r4IP,, AP,, ?r4S3

1 qs, u

71/2 N 1r4I P -.71/2 (ax).! pull 10 II 71/2 1r4IP2 -j Z/2

Here [1,1] * = 0 is trivial. In fact q* [1,1] * = [q, q] = 0 is trivial since S3 is an H-space and (X)*[1,1]*[1,11*(1XAX)* is trivial since 0=[1,1]* for n=2.

(11.5.16) Definition of the generatorsLet k, n, m be powers of primes. The canonical generator

X = X,,k E Hom(Z/k, Z/n) = Z/k *71/n carries 1 to n/(k, n) where (k, n) is the greatest common divisor. Let

(1) be given by B2 in Theorem 1.4.4 so that X,k induces X in homology. In the following the context always shows whether X,k denotes a homomorphism or a homotopy class. We choose S 2 E ir4 IP2 as in Theorem 11.5.13 and we set

1r4YP,,. (2) 370 11 HOMOTOPY GROUPS IN DIMENSION 4

Then µ(4n) E FT(71/n) =71/2*71/n is the generator. Moreover let 62,2= 72(62) as in Theorem 11.5.13. For (n, m) not equal to (2,2) letSn m E 1r4IP,, A Pm be the unique element for which µ(fn m) E 71/n *71/m is the canonical generator and for which (X2#x2)*bn,m=0' (3)

See (11.5.10) (1) for the definition of x2 #X2.Here Xz #X2 induces a pull-back diagram between elementary exact sequences (11.5.4) (2) so that Snmis well defined sincef'jk H4 Xz #X m = X * X = 0 for (n, m) not equal to (2, 2). Using fn,m wedefine m =fnm qas in Theorem 11.5.9. Next we obtain ?lnmE[Y-2Pk,IP.APM 1by

in#Xm k if m = (n, m) 71n,m = (4) - (Y.T21) * rlm, n otherwise.

Here in: S2 c lPn is the inclusion and we use the product # in (11.5.10)(1). One readily checks that µ(n, m) E Hom(l/k, 71/n ® 71/m) is the canonical generator. In (4) the numbers n, m or k may also be 0. We define yk E [y 2Pk, Y.Pn], n even, by setting

yk = y, "(1X2n) (5) where yen is the following generalized Hopf map; see Definition 11.2.2. The element ye" E [VP2n, F,Pn] is the unique element which is represented by a twisted map; see (11.4.4), associated with the diagram

S3 I''''21 S2 V S2

2nI (6) S3 n2 and which satisfies

)'2(y2n) = 7Inn - Sn?n (7) Here y2is the James-Hopf invariant. Using (6) we see that %2"is a generalized Hopf map and µ(yk) E Hom(71/k, I (71/n)) with r(71/n) = 71/2n is the canonical generator. If n is odd let P = 2:1/n - 71/n be the isomorphism which is multiplication by 2. This isomorphism yields the homotopy equivalence B3(2). I2Pn = E2Pn the homotopy inverse of which is B3(1/2); see Corollary 1.4.6. Then we set

Yn = [i,, 1]B3(1/2): E2Pn-Y..P,, (8) 11 HOMOTOPY GROUPS IN DIMENSION 4 371 where [i",1]: IS' A P" -> I,Pnis the Whitehead product of the inclusion i": I,S' c IP" and the identity 1 of IP". Again y,, is a generalized Hopf map and yk = yields the canonical generator µ(yk). Compare the proof of Lemma 11.5.19 below. For n odd we obtain the James-Hopf invariant of yn by the formula

72(yn) = -1n,n - Sn,,B3(1/2). (9)

This follows from (A.10.2)(h) and (11.5.20)(1) below.

(11.5.17) Lemma The generalized Hopf map yenis well defined by the conditions in (11.5.16)(6), (7).

Proof Using the exact sequence (11.2.5X3) and its cross-effect sequence we obtain the following commutative diagram; for this recall that L3(A, 1) = 0 if A = i/n is cyclic. Let k, n be powers of 2.

Ext(7L/k,7r3(lP")®7L/2)/K)-a [1ZPk1YP"] A 0 FT,(71/k,7L/n)

a1(Y201). 1r2 Ih Ext(7L/k,ar3(EP"nP")(9 A V[dZ1k,A0d,]

CIX II AP"]

(1)

Here we have K a i/2 if k = n = 2 and K = 0 otherwise. The map h *is induced by h in (11.1.4) and C,, denotes cellular chains. Since (y2 (9 1)* is an isomorphism this diagram is a pull-back. Now let k = 2n. Then Addendum 11.4.5 shows that d(y2n) is the chain map associated with diagram 11.5.16 and one can check

A(, n- SnZn) =h,,A(y2n). (2) For this we consider chain maps F = (FI, FO)

7L ,Z ®7L/n 2n1 In®1-0 (3) 7 - - Z0i/n

Then h * k(y2n) is represented by F = (1, 1) and ) is represented by F = (1, 2) and A(6 ?",) is represented by F = (0, 1). This proves (2) and therefore the pull-back (1) shows that d,2," is well defined. 0 372 11 HOMOTOPY GROUPS IN DIMENSION 4

Next we consider the James-Hopf invariants

y2: ir4(Y.Pn) - 1T4(IP A Pn)

Y2: [i2p,, I.Pn AP }.

They satisfy on generators the following equations:

(11.5.18) Lemma Let k, n be powers of 2. Then

y2(en) = en,n

Y2(ER) = enn y2(fn) _ (n/2)4n,n y2(Sn) _ (n/2)n,n 2(n, k) k yz(y,_ " n. (2n,k)°'n (2n,k)

Proof We firstcheck theformula ony2(6n). We have y2Qn)_ 62 = (X,2#Xn)*Y2 62 = 2,2Hence we get for n > 2

(X2#X2)*Y2Q)=0 since for n > 2 we have Xz X = 0. On the other hand, µy2( fn) = n/2 E 7L/n *7L/n = 7L/n since for the canonical generator X E Hom(7L/2,7L/n) we have X(1) = n/2 and hence

l Y2(Sn)=i (Xn#Xn)*C2,2 =(X*X)µ(52,2) with µG2,2) = 1 and X * X = X Next we check the formula on y2(y) We have

yz(Yk) = y2(y2n )(EX2n) Wren .X2 _ Sn2n`yn X12n ) and now we can use Lemma 11.5.24(f). This yields the result since ez 2=0' 0 11 HOMOTOPY GROUPS IN DIMENSION 4 373

In addition to Lemma 11.5.15 we now consider for the Whitehead square [1, 1] the induced homomorphism

[1,1]*: [12Pk,1Pn AP ] , [c2Pk,1Pn]

On generators we get the formulas:

(11.5.19) LemmaLet k and n be powers of 2. Then

[1,1]*errn n=0

[1,1]*Srtkrt=0 (2n,k) k kk 1]#stn.n (n,k)Y. +tSnen.

Here let S,k = 1 for n = 4, k E {2, 4), and S,k = 0 otherwise. In particular we get for the generalized Hopf map yen the formula

2,Y2n = [1, 1]7In,n

Proof The first two equations are consequences of Lemma 11.5.15. Since -q, ,k,,, = in#X we have [1,1] *7irtn =[in,1] (1 A X,k) where in: S2 C EPn is the inclusion. Here the Whitehead product [in,1] E [ES' A Pn, EPn] is a twisted map associated with the right-hand square of the diagram

S3 k/(k,n) [i1,i21 S3 S2 V S2

kI I(n,1)

n/(k n) 2n2 S3 I S3 S2

The diagramrepresentsA([in, 1](1 A X,k)) E I'T#(71/k, Z/n).Using (11.5.16)(5), (6) we therefore have

(2n, k) A([1,111n,n)= A(yk). (n, k)

Using the pull-back diagram in Lemma 11.5.17(1) it remains to show that the James-Hopf invariants coincide. By Lemma 11.5.18 we have

(2n,k) (2n, k) - k k k\ Y2(ynk ) = f,,.n Yz (n, k) Yn (n, k) (n, k) 374 11 HOMOTOPY GROUPS IN DIMENSION 4

On the other hand, we obtain by (11.5.10X9)

Y2(11,1= (Y2[1,11)Yln.n

k k kenk 2 71n.n tk' _ IT21 . - + (n,k)Snn + Srt' where in the last equation we have used the next result (Lemma 11.5.20).

(115.20) Lemma Let k, n, m be powers of 2. Then the interchange map T21 on Pn A Pm induces the isomorphism - (IT21)*: [y2Pk,I.,Pn APm] [I.2Pk,IPm AP. which on generators is given by the following equations:

(I T21)*Enkm-Emkn

T21)*Snm -Sm,n

k (ET21 ) = 74,n for m not equal to n

k k ttk k k in,rt=-?7n,n+(n,k)Sn,n+snen

Here let 6,k = 1 for n = 4 and k E (2,4) and S = 0 otherwise.

Proof The first two equations follow from Lemma 11.5.15 and the third equation is a consequence of Definition (11.5.16)(4). It remains to prove the fourth equation. One readily checks that 'l: [j2P, I.P, A P,, ] -* A ® dA] with A = 2/n carries both sides of the equation

(1 Rn+SnEn,n (1) to the same element. Hence the exact sequence in the bottom row of Lemma 11.5.17(1) shows that for appropriate Sn E Z equation (1) holds. We set S2 = 0 since E22,2 =0. Since

-qn T1n,n(IXR) (2) by (11.5.16X4) and since

k (3) Cn, Xn) = (n, k) rt we see that (1) implies SR = Sn'k/(n, k) by a similar argument as in (3) where 11 HOMOTOPY GROUPS IN DIMENSION 4 375 we replace 6 by e. We have to compute Sn in (1) for n > 2. For this we compute in [ I2P,,, I P2 A P2 ]

(XZ#Xn)*(1 + (4) =(1+IT)*(22*nn,n

= (1 + IT) * (5)

= 22(Y- X2) = 622(gI X2) (6) = (n/2)C2 2 = (n/4)e2,2forn > 2.

Here (5) is a consequence of the definition of qnnand (6) follows from (1) since ez 2 = 0. On the other hand, we get by (1), (4)X2#X2")*(A, n+snen,n) X2#X2)*(4,nq)+sne2,2rt n n

= sne2,2 (7) Here (7) is a consequence of (11.5.16)(3). Hence we have proved that 5,, = n/4 modulo 2 for n > 2 since ez2 0 for n > 2. Hence we get S nk/4(n,k)forn>2and 52 =0.

(11.5.21) Lemma Let n, m be powers of 2 and let mm: y' 2m = E S'/ Pm c P n/ P m be the inclusion given by S' cPn. Then we have in [E2Pm, EP, APm]

_f"'7n m ifmsn inm 17rt,m + 6n mm if m > n.

Proof We use the short exact sequence, A = 7L/n, B = 7L/m

Ext(71/k,ir3(YPrtAPm)®7L/2)/K»[y2Pk,Y.PnA Pm]] [dz/k,d+®B]

C. II [C*1,Pk,C*IPf APm] where K = 7L/2 if k = n = m = 2 and K = 0 otherwise. For A = B this is the bottom row in Lemma 11.5.17(1). Let k = m. By definition of i7nm the equation in (11.5.21) is true if m = (n, m); see (11.5.16)(4) where X, is the identity. Now assume m > n. Then we have by (11.5.16)(4)

' 7n,m =-($T21)*77m,n = (1) 376 11 HOMOTOPY GROUPS IN DIMENSION 4 where ,q,,= y2P C Y.PA P,,is the inclusion. One can check that A above satisfies AU,,) = A(rlrt+ 6 , , m ). Moreover we have

( Xn #Xn )inm = 1n 7nn. (2) On the other hand, we have by (1)

(Xn#Xn )('nom

Xn #Xn )tm (4Xn ) +r( Xn #Xn )Snm (3)

_ -(Y-T21)in n(1Xn)+.Z'nr#Xn )Snmm

_ n+(Xn#Xn )Snm (4)

gm 71n n - (m,M n) Cm. + (Xn #Xn ) m (5)

1nn (6) In (4) we use (2) and in (5) we use Lemma 11.5.20, Moreover (6) is a consequence of Lemma 11.5.24(b) below since N( Xn * X,,)m/(m, n) = m/n for m > n. Since X,"#X induces an isomorphism for the kernel of A above this completes the proof of Lemma 11.5.21.

For cp E Hom(Z/n, Z/m) let (11.5.22) E be defined by Theorem 1.4.4. The context below always shows clearly whether qp denotes a homomorphism or a homotopy class. Recall that B2(q') is the suspension of a principal map P -* Pm inducing gyp. Now let m, n, r, s, k, t be powers of 2 and let p E Hom(Z/n, i/s), ip E Hom(l/m,71/r), and r c- Hom(71/t,77/k). Then we have induced homomorphisms

(a) rp*74 Y- Pn -> 741P, (b) cp#1(r)*: 7r4EP APm - 1r4 Y. P5 APr (c) P*: [I2Pk,IP1

(d) [y2Pk,lPnAP. - .2Pk,Y. P., APrJ

(e) (IT)*: [E2Pk,y P

(f) (Y.T)*: [F,2Pk,IP APm] - A These homomorphisms are computed on generators by the following result. For this we use the notation: 11 HOMOTOPY GROUPS IN DIMENSION 4 377

(11.5.23) DefinitionLet A = (7L/n)x and B = (7L/m)y be cyclic groups with fixed generators x and y respectively and let cp a Hom(A, B). We choose a number N(ip) which satisfiesq (x) = N(cp)y and let N(ip) be the unique number with 0:!5 N(cp) < m and p(x) =Rip) y. Moreover we define M((p) by nN(rp) = mM(gp). For example 7L/n,7L/n ® Z/m, and 7L/n * 1/m are cyclic groups with canonical fixed generators.

(11.5.24) Lemma The homomorphisms of Lemma 11.5.21 satisfy the following equations (a)-(f).

(a) ,p. en = N(cp) es P* nM(q)M(0 6s.

(b) =N(tp(9 +I)es r

(cp#G)*Sn,m=N(cp*&Jr)s.r

(c) cp en =N(ap)es (P*4 =M(0M(,p)fsk. (2s, k)

(p*Yk = N(cp)M(,p) ysk. (2n, k)

(d) (cp#,i)* e,km =N((p(9 +/)e5r

((p#,p)* fn m =N((p * tp) &kr

r+ss,rSsrr)1(4'Xm) form n.

Here we set e3= 1 for r > s and 55 r = 0 for r >_ s. The right-hand side can be computed by the formulas in (f) below.

(e) (Y.T)*ee, =M(T)e'

(Y.T) *Ck =M(T) fn (t, 2n) (>T)*.k=N(T)(k,2n)

(f) (!T)*errn,m =M(T)en.m

(1T)*Sn,m =M(T) (t,a) 5,,"(N(7-)(N(T) (ET)*71,k,m =N(T)(k, a) fn'm+ 378 11 HOMOTOPY GROUPS IN DIMENSION 4

Here we set a = min(m, n) and we set Sa ` = 1 for t = a = 2 and k < 4 and So , ` = 0 otherwise.

Proof All formulas for --terms are clear since B2(gp) has degree N(cp) on the bottom sphere S2 and has degree M(cp) on the 3-cell. Moreover one readily checks the equation

cPX,, =Xs2(M(cp)Xz) for homomorphisms. For a = M((p) we have in [Y.P2,1P21 the equations; see Theorem 1.4.8 a(a - 1) B2(ax2) =aX2+ 2 'g24

= aX22 +a(a - 1)X2 2= a2Xz where we use (11.5.12). Since B2 is a functor we thus obtain the second formula of (a). Now we prove the second formula of (b). The operator µ carries both sides of the formula to the same element. Next we obtain the coefficient of es,by applying X2#X2; see (11.5.16) (2). We have X2 cP = (P2 X2 with P2 = cp ® 71/2 in Ab and hence we get the element

)&mCC which is trivial for (n, m) not equal to (2,2) and which is X2,2 for (n, m) = (2, 2). Here we have P2#412 = 0 for (s, r) not equal to (2,2) and (n, m) = (2,2) since then cp2 = 0 or r(r2 = 0. On the other hand, the element ( X2#XZXN(rp * alr)S ,)istrivialfor(s, r) not equal to (2,2) and is 2 for (s, r) = (2, 2). For (s, r) = (2,2) however we get p * 41 = 0 for (n, m) not equal to (2, 2). This completes the proof of the second formula of (b). It is enough to prove the third equation in (c) for k = 2n since we can apply (e) for k not equal to 2n. Hence let k = 2n. One readily checks for A in the top row of Lemma 11.5.17(1) the equation

A( p*Yn1 n) = P* A(y2n) = aA('Yt2n) with a = N(v)2(n, s)/s. Using the pull-back diagram of Lemma 11.5.17(1) it hence remains to show that James-Hopf invariants satisfy

Yn") = aY2(Y:2n). On the one hand, we have

Y2('P*Yn2n) _ n -?n) n) = N(W) rlssY.(9Xn - N((p *,)C,,2; C,J2; 11 HOMOTOPY GROUPS IN DIMENSION 4 379

Here we use (d). For P = (3Xzs with p = N( cXn, s)/s we have (Pyn " = XS2S BXis) in Ab and one checks as in the proof of (e) that _ zs Hence we get

Y2((p*Y2") = (p)e,2 with (3N((p) = a. On the other hand, we have

y2(ays2") = a(ys2siXzs)

zs 2s 2n 'S 2s = a(nssS1Xs2s)1Xis -N((p)M(9)es2,

Here we use (f) and the definition of 77s?s. Hence it remains to show

(N((p)M(cp) -.N(qp * (p))es2s = 0. (*) Here we have N((p * gyp) = N((p)M(g) modulo s, hence (*) holds for s > 2 since then 2 Ss2s = 0. For s = 2 and tp not equal to 0 we have N(tp) = 1, M((p) = n/2, and N((p * gyp) = 0 for n > 2. Hence (*) holds for s > 2, n > 2. Now it is clear that (*) holds for s = n = 2. This completes the proof of (c). For the proof of (d) we may assume m = (n, m); see (11.5.16)(4). Then we obtain by the composite in the top row of the commutative diagram

12P, Y.Ps A Pr IJ.l

Here in,mand is, arethe canonical inclusions. Let 8,, =8 = 0 if r < s and let 6 = 1 for r > s. Then Lemma 11.5.21 and the diagram show

=N(W)"(Tla This yields the equations in (d). For the proof of (e) we only consider (Yr)*yk. We have

(I T)*Yk = y2n(y y2n.( X2nT) where X2"T = aX2n with a = N(TXt, 2n)/(k, 2n). Moreover we have by Theorem 1.4.8

t 2n 2 1(aX2") = a(Y-Xzn) + (2) i1139 2((t2n)) 380 11 HOMOTOPY GROUPS IN DIMENSION 4

Here the q3-term vanishes for t > 2 since 2713 = 0. Moreover for t = 2 we get (;)n2j3q= 0. This shows that actually Y.(aX2) = a'XZn and the third formula of (e) is proven. For the proof of (f) we only consider the element (Y.T)*ri m for m 5 n. Then we have

(I T)*71n.m m(Y' )G (yT))

=1n my.(XmT) Now X,7- = ain Ab with a = N(T)(t, m)/(k, m). Therefore we get by Theorem 1.4.8.

a(a-1)t XmT)=aIxm'+ 2 -2NZ'rl3q with N = N( m/(t, m). This shows (1T)*71,km =a7ln,m +Bsmwhere

/3=(mt/2(t,m))(a(a-1)/2)modulo2.

Hence we get 1N(r)(N(T)-1)/2fort=m=2,k2! 4 0 otherwise

Next we consider the pinch map q: IPn - S3 which induces

q*: 'zr4YPn -* 17-4S'= (71/2)13

q*[jZPk,EP"I -, [2Pk,S3] =71/(k,2)rl3q

(11.5.25) Lemma q*en=0 q* (n/2)773 q*ek=0

q6,k = qYk = 0. 11 HOMOTOPY GROUPS IN DIMENSION 4 381

Proof We have q. ?;2 = r73by Theorem 11.5.13 and hence q. & = q* X where Moreover q*yk=0 by (11.5.16) (6).

The reduced diagonal A: P, --> P, A Pn satisfies

(11.5.26) A = (n(n - 1)/2)iq: Pn - S2 --> Pn A P, Therefore we can apply Lemma 11.5.25 for the computation of (IA)* on 7r4XP,, and (E2Pk, IP,,]. Moreover we have to compute the induced maps Y.(1AA)*:7r4(IPrtnP,,)-ir4(Y-P, APAP,n)7L/n®7L/m I(i AA),: [y 2Pk,I Pn nP] [y2pk,I Pn nPm nPml = 7L/k ®7L/n ®7L/m and similarly(A A 1)* where A is the reduced diagonal on Pm and Prt respectively. Let 1,, E 7L/n be the canonical generator.

(11.5.27) LemmaLet k, m, n be powers of 2. Then E(1 A A)*and VA A 1)* cany the elements em,n, ekCm,n, sm,n (m not equal to n) to 0. Moreover

X,(1 A A),Sn.rt =Y.(A A 1)* Sn,n = (n/2)1® ® 1,, X(1 A A)* e, ,k =I(AA 1),e Snkn = (n/2)lk ® 1® ® In nk VA A I) *7)n,m = 5,,,,n2(kn)lk ®ln ®lm mk (1 A ' * 1 7 , k,, lk ® In ® lm ) = (1 - Sn m)2(k m) where 8n,m =1 form > n and Sn,m = 0form 5 n. Proof We have the commutative diagram 7r4(Y.PnAPm) ha H4y.1PnAPm =7L/n*7L/m (FqA 1).1 n 1 7r4(ES2APm) = H4(FS2APm)= 7L/m where h is the Hurewicz map and q is the pinch map. Hence we obtain VA A 1) * fin,m by (11.5.26) since h(fn, m) is the canonical generator. This shows that the composite

l/n * 7L/m c 7L/mom7L/m -. 7L/n 0 7L/m 382 11 HOMOTOPY GROUPS IN DIMENSION 4 carries the canonical generator to X,(0 A 1)*Sn.m.This yields the result on nm andsNext we consider qm for m < n so that by(11.5.26) X(0n1)*fl,'m=0

1(1 A m =(m/2)M( X,f)lk 0 In ®lm since 71rtm =i,,.mY.Xnk; see Lemma 11.5.21. For in > n we use (11.5.16)(4).

(11.5.28) Remark Let A and B be direct sums of cyclic groups and for cp E Hom(A, B) let sip: M(A, 2) --> M(B, 2) be a map which induces rp. Then the formulas in this section and in Section A.10 allow explicit computations of the induced maps

(s(p) :ir4 M(A, 2) - ir4 M(B, 2)

(sip)*: [E2pk,M(A,2)] [Y2Pk,M(A,2)] on generators in Remark 11.5.3 and Theorem 11.5.9. For this we need, in particular, the left distributivity law of Theorem A.10.2(b).

11.6 The suspension of elementary homotopy groups in dimension 4

Let X and Y be pointed spaces. We say that the set of homotopy classes [ X, Y ] is stable if the suspension yields bijections for n >_ 1

4: [In-1X, Cn-1Y] = [G"X, 4nY]. For example we get

(11.6.1) PropositionThe groups 7r4I Pn, ir4F,Pn n Pm, [EZPk, Y-Pn A Pm ], and [ VPk, VPn ] are stable for k, n, m >_ 0.

We want to describe the suspension on elementary homotopy groups in dimension 4. By Proposition 11.6.1 we only have to consider

(11.6.2) 1: [I2Pk,y PnI - [13Pk,F2Pn].

For this we define generators in [E3Pk,12Pj as follows.

(11.63) DefinitionLet k, n be powers of 2 and r >_ 2. Then e, = e is the composite . Sr -a jr-1Pn e,k : sr+ 2' 11 HOMOTOPY GROUPS IN DIMENSION 4 383 where q is the pinch map and i is the inclusion. We also set e = i77, and ek = 77r2q. Now we choose generators 1P, ] 2 E [ S°, = 77/4, see Theorem 11.5.13 and

7772E [13P2,S3J =71/4 which are stably Spanier-Whitehead dual to each other. Then we obtain the composites

yr 2 #2.Jr-Ip2 xR ck: irpk g .Sr+2 Iripn r>2, jr- 2 rp2 Sr 71n . y 'Pk r-'P,,, r> 3, - and we set,, =X'(I'2) and rlk = (y' '712 Here X = B,(X) is given by the canonical generator )r = X,, E Hom(//k,71/n); see Corollary 1.4.6. We also write e,k = e, (r), 6,k = 4k (r), and 7),k = 77n (r) if we want to specify the dimension r. Hence ek(r) and 6,k(r) are just suspensions of the corresponding elements in Theorem 11.5.9 and (11.5.16). The element 77,k(r), r3, however, is a new type of element which is Spanier-Whitehead dual to fa.If k or n is odd we have [I3Pk, 0 for k, n >_ 0. If n = 0 we get the Spanier-Whitehead dual of [S°, VPk] in Theorem 11.5.9 given by

[13PkS3J (77/4)77 with 277 = ek = 2 (11.6.4) (77/2)77 ED (71/2)e k = 2` >_ 4 where 77 = 77k and e = ek. Moreover the suspension

I: [y2Pk,S2] =77/(k,2)772q-' [13Pk>S3] carries the generator 77z4 to ek and hence 77kis not in the image of X. Moreover we get

(11.6.5) TheoremLet k and n be powers of 2. Then C = f ,,k,e=en,n=77 n satisfy

(77/2)x® (71/2)77 with e= 0, (k, n) = (2,2) (Z/4) f ® (77/2)7) with e = 2 6,(k, n) = (> 4,2) (71/2) f ® (77/4)71 with e = 277,(k, n) _ (2, >_ 4) (71/2)C® (Z/2) e (71/2)e, (k,n) = (z 4, z 4). 384 11 HOMOTOPY GROUPS IN DIMENSION 4

(11.6.6) TheoremLet k and n be powers of 2. Then the suspension .: {y2Pk,Pn} - carries e, to e, and R to and satisfies

0 k__

1(yk)_ ? + 6 e z (k,n)=(4,2),SE{0,1}

11'k otherwise.

We do not know whether S = 0 or S = 1 for (k, n) = (4, 2). Hence for k > n the suspension is surjective while for k< n the suspension in Theorem 11.6.6 is not surjective.

Proof of Theorems 11.6.5 and 11.6.6The definition of q,,k shows that the diagram k j3Pk y2Pn it 1i

773 S4 S3 homotopy commutes where 173 is the Hopf element. This follows by duality from q1=2 = iin Theorem 11.5.13. Hence the operator

[ y3Pk,I2P, ] -* Hom(Z/k, 7r4Y.2P,,) = 7L/k * (7L/n (&71/2) carriesq1,to the generator. Hence using stability, Theorem 11.6.5 is a consequence of Theorem 8.2.10; compare also Theorem 9.2.7. Next we prove the formula for Yyk in Theorem 11.6.6. For k:!5; n we obtain £yk = 0 by Lemma 11.5.19. We now consider jy2n Here Yy2rtis a principal map associated with s4 0a S3 2n1 I. 113 S4 S3 This implies that Iy2n = with S" E (0,1). Since (X2n)*en" = 0 for k > 2n we hence obtain Iyk = % for k > 2n. Next we get

n 2rt= 2, 2n X2 712 X2 en2' =- e2

n2n = y2n X2 yn 2 by 11.5.24. This implies for n > 2 2n = (n)( 2n _2n\ 6n e2 \ X2 yn Tin 1 =ly2n-n2n=0 so that 6n=0forn>2. 12 ON THE HOMOTOPY CLASSIFICATION OF SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

The classical result of J.H.C. Whitehead on simply connected 4-dimensional homotopy types relies on the computation of the group I'3 X which fortu- nately has the simple description r3x=r(7r2x) in terms of the r-functor. A homomorphism r(A) - B is given by a quadratic function 77: A --+ B which is the algebraic equivalent of a simply connected 3-type, denoted by K(i1, 2). The homotopy classes of maps K(i7,2) -->K(r/',2), however, do not coincide with the obvious algebraic maps r/ -s 77' between quadratic functions in the category lAb. In fact, the homotopy category types'' of simply connected 3-types is a complicated linear extension of the category FAb. We therefore introduce the diagram of functors types' G- rAb(C)

rAb where FAb(C) is an algebraic category which via G is a better approximation of the category types2 than rAb. Our classification of simply connected 5-dimensional homotopy types X relies on the computation of the group

r4(X) = r,K01, 2) = I',(71) as a functor in X or in 71 = Tjx E FAb(C). The algebra needed to describe the functor r4 is somewhat bizarre. Given the functorr4 and also the bifunctor r3 with r3(H, K(77,2)) = r3(H, 7) we are able to describe algebraic models of simply connected 5-dimensional homotopy types X for which H2 X is finitely generated. We also consider such homotopy types for which H2 X is uniquely 2-divisible or free abelian. 386 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 12.1 The groups G(q, A)

Recall that Pq = S' u9 e2 is the pseudo-projective plane of degree q which yields the Moore space M(7L/q, n) _ I"-'Pa, n >_ 2. Let X be a space. In this section and in Section 12.2 we consider the homotopy group (12.1.1) it"(7L/q,X)=[En lp" X] with coefficients in the cyclic group 7L/q. Let f Cyc be the full subcategory of Ab consisting of finite cyclic groups. Then (12.1.1) yields the functor ir"(-,X):fCyc°P_*Gr (1) which carries 7L/q to the group (12.1.1) and which carries a homomorphism gyp: Z/q -* 71/t to the induced homomorphism cp* = (B"cp)*: 7r"(71/t,X) --> ir"(l/q,X) (2) given by the functor B" in Corollary 1.4.6. The group ir2(7L/q, X) in general is not abelian; the universal coefficient sequence, however, yields the central extension of groups (n > 2)

(12.1.2) Ext(Z/q, ir, ,X) >* 7t"(71/q, X) -µ' Hom(7L/q,1r"X) which is natural in 7L/q. We use the following notation. For a small category C a C-group is the same as the functor C°P -- Gr. Let C-groups be the category of such functors; morphism are natural transformations. Hence (12.1.2) is a central extension of fCyc-groups. Let S" V"-'Pq q-I S"+' be the inclusion and pinch map respectively. Then A in (12.1.2) carries 1 ®x E 71/q ®7r"+, X = Ext(7L/q, a"+ 1X) to A(1 x) = q*+,(x). Moreover p. in (12.1.2) carries an element y e 7r"(71/q, X) to (i")*y E 7L/q * it"X = Hom(7L/q, 1r"X).Commutatorsinthegroup ir2(7L/q, Z) satisfy the following rule

(12.13) PropositionFor x, y E ir2(71/q, X) we have the formula -x -y + x + y = (q(q - 1)/2)qs*[i*,izy] where [ - , - ]: 7r, X ®a2 X - ir3 X denotes the Whitehead product.

This result is originally due to Barratt [TG].

Proof of Proposition 12.1.3For pointed CW-complexes A, B let A A B = A X B/A V B be the smash product and let A: A -+A AA be the reduced diagonal. The commutator satisfies the formula -x-y+x+y=(Y.A)*[x,y] 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 387 where [x, y] E [1Pq A Pq, X] is the (generalized) Whitehead product; compare Baues [CC]. By (III.D.20) in Baues [CH] we know that the reduced diagonal 0 of P. is part of the homotopy commutative diagram

P9I -° > Pq A Pq

q2 l li,A i, S2 --.S'AS'=S2 Here q2 is the pinch map and i iis the inclusion and v is a map of degree q(q - 1)/2. This yields the result.

We now introduce a purely algebraic construction of a group denoted by G(q, A). For this we use the properties of Whitehead's functor F in Section 1.2. The topological meaning of the group G(q, A) is described in Theorem 12.1.6 below.

(12.1.4) DefinitionLet A be an abelian group. We have the natural homomorphism between 1/2-vector spaces

H: F(A) ® 71/2 = F(A ®7L/2) ® 71/2 - ®2(A ® 7L/2) (1) with H(y(a) ® 1) = (a ® 1) ® (a ® 1). This homomorphism is injective and hence admits a retraction homomorphism

r: ®2(A ® 71/2) -> F(A) ® 71/2 (2) with rH = id. For example, given a basis E of the 71/2-vector space A ® 7/2 and a well ordering < on E we can define a retraction r on basis elements by the formula (b, b' E E) y(b)®1 forb=b' r(b (&b') _[b, b'] ®1forb > b' (3) 0 forb < b'.

Now let q >_ 1 and let

JA: Hom(71/q, A) =A *7L/q cA-P.A ® Z/2 (4) be given by the projection p with p(x) =x ® 1. Also let

PA: F(A) ® 7/2-P F(A)® Z/2 ® 7/q = Ext(Z/2 ® 71/q, F(A))

P + Ext(7L/q, F(A))(5) 388 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA be defined by the indicated projections p. Then we obtain the homomorphism J A,: Hom(7L/q, A) ® Hom(Z/q, A) --Ext(7L/q, TA) (6) Ar =pAr(JA OiA) which depends on the choice of the retraction r in (2). Clearly Ar is not natural in A since r cannot be chosen to be natural. However one can easily check that Ar is natural for homomorphisms q :7L/q -> 7L/t between cyclic groups, that is

®(p*)= P*L1r. (7) We now define a group Gr(q, A) = Hom(7L/q, A) X Ext(Z/q, f(A)) (8) where the group law on the right-hand side is given by the cocycle tr, that is (a, b) + (a', b') = (a + a', b + b' + Ar(a ®a')). (9) This yields a functor

G,(-, A):fCyc°P_3Gr (10) which carries Z/q to Gr(q, A). For w:7L/q -> 7L/t we define 4P*: Gr(t, A) - Gr(q, A) by v* = A) x Ext((p, I'(A)). It is clear that v* is a group homomorphism. In addition we get a central extension

Ext(7L/q, I'(A)) -°--+ G,(q, A) -' Hom(7L/q, A) with ti(b) = (0, b) and µ(a, b) = a. This extension is natural in 7L/q so that Gr( -,A) is a central extension in the functor category f Cyc-groups in the same way as in (12.1.2). The next result shows that the extension Gr(-, A) does not depend on the choice of the retraction r.

(12.1.5) Lemma For two retractions r, r2 in Definition 12.1.4(2) there is an isomorphism of groups X:Gr(q,A)=G,,(q,A) which is natural in 7L/q and which is compatible with 0 andp., that is XO = and µX =,u. We therefore omit r and write G(q, A)= Gr(q, A).

The lemma is also a consequence of Theorem 12.1.6below. Since the proof of this theorem is rather sophisticatedwe first give an independent algebraic proof of the lemma. 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 389

Proof of Lemma 12.1.5Let v: ®2(A ® 7L/2) -- A2(A 0 71/2) be the quotient map for the exterior square. Then the retractionsr,, r2yield a unique homomorphism

m:A2(A®7L/2)-*1'(A)®71/2 withmv=r2-r1. (1) Let the homomorphism

6: f(A (& 71/2) --> A2(A 0 Z/2) (2) be defined on a 7L/2-basis E of A 0 7/2, namely Bye = 0 and 6[e, e'] = e A e' for e, e' E E. One readily checks that B is well defined and that B[a,b]=anbforalla,bEA®71/2. We obtain by B the quadratic function BA =PAmSyjA: Hom(71/q, A) - Ext(7/q, F(A)) (3) where we use pA and jA in Definition 12.1.4. We again observe similarly as in Definition 12.1.4(7) that BA is actually natural in 71/q. We use BA for the definition of the isomorphism X in Lemma 12.1.5. We define the bijection X: Hom(71/q, A) X Ext(71/q, I'A) -* Hom(71/q, A) X Ext(7L/q, rA) (4) by X(a, b) = (a, b + SA(a)). Thus also X is natural in 71/q and X is an isomorphism of groups (see Definition 12.1.4(9)) since we have Ar(a, a') + BA(a +a') -pAri(jAa (&jAa') +PAm By(jAa +jAa')

=pA(r1 JAa (&IAa' + m6[jAa,jAa']) + BA(a) + BA(a')

= pA(r, + mv)QA a (& jAa') + B,(a) + BA (a')

= (PA r2jA (& jA)(a (9 a') + B,(a) + BA W)

= Ar,(a, a') + 3A(a) + BA (a'). This completes the proof of Lemma 12.1.5.

The next result shows that the algebraically defined group G(q, A) in Lemma 12.1.5 is isomorphic to a homotopy group of a Moore space.

(12.1.6) TheoremThere is an isomorphism of groups X : G(q, A) = ire (7L/q, M(A, 2)) which is natural in 7L/q and which is compatible with 0 and µ, that is XD = A and AX = t..,. 390 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA Thus we can choose for each abelian group A a retraction r and an isomorphism X as in the theorem. We will use this isomorphism as an identification. The group G(q, A) in the theorem yields a purely algebraic description of the homotopy group ir2(7L/q,M(A,2)). The suspension j;" - 2,n > 3, induces a push-out diagram of groups

Ext(Z/q, F(A)) '7r2(Z/q, M(A,2))-µHom(Z/q, A) ly, (*) I-. 2 II Ext(7L/q, A (9 Z/2) - ir"(Z/q, M(A, n)) -µ* Hom(Z/q, A)

Hence we can apply the theorem also for the computation of the group 1T,(7L/q, M(A, n)), n > 3.Inparticular,acocycleforthegroup ir"(7L/q, M(A, n)) iscr * 0,. The proof of Theorem 12.1.6 has several parts. We first consider commutators.

(12.1.7) Lemma The commutator for x, y e G,(q, A) satisfies the same formula as the commutator for x, y e ir2(77/q, M(A, 2)), namely -x - y +x + y = (q(q - 1)/2)0([ Ax, Ay] ® 1).

Proof For x = (a, b) and y = (a', b') we get the commutator formula in G,(q, A) as follows: -x - y +x + y = O(O,(a, a') - O,(a', a)) _ APAr(JA (& IA)(a ®a' +a' (9 a)

= APArH[JAa,JAa'] = OPA[IAa,.IAa'] = (q(q - 1)/2)A([a, a'] (9 1).

Next we consider a splitting function s = sq for µ:

(12.1.8) (Z/q, M(A, 2)) 4 Hom(7L/q, A), that is µs = id. Such a splitting function yields the cocycle A,: Hom(Z/q, A) X Hom(Z/q, A) ---' Ext(7L/q, FA) with A,(x,y)=0-`(s(x+y)-s(x)-s(y)). We say that s=sq is natural in Z/q if for cp e Hom(Z/q,7L/t) we have sq cp* = B2(0*s1. Clearly for natural splitting functions also the cocycle A, is natural in Z/q.

(12.1.9) PropositionLet A be a direct sum of cyclic groups. Then there exist splitting functions sQ, q > 1, which are natural in 7L/q. 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 391

Proof Let A = ®, (71/a,)ai with i E I, a; >- 0, and let < be an ordering of I. Then x (=- Hom(Z/q, A) is given by coordinates x; E Hom(l/q,71/a,) with xi non-trivial for only finitely many indices i. Hence we can define

sq(x) _ E a.B2(x,) lEt where the sum is the ordered sum in the group [lP9,M(A,2)] and where ai: M(71/a 2) -- M(A, 2) is the inclusion; see (1.5.2). We clearly have for rp:7L/t -> Z/q B2((P)*sq(x) _ aiB2(x,)B2(c')

_ =syW*(x) iEI where (xi cp) = (x (c), is the coordinate of xc'. In the first equation we use the fact that B2 is a suspended map and in the second equation we use the functorial property of B,. Proof of Theorem 121.6 We first assume that A is a direct sum of cyclic groups (Z/a,)a i E I, as in the proof of Proposition (12.1.9). For the splitting function s = s9 (which is natural in Z/q) in this proof we get the cocycle A, by z5(x,y) =Sq(x+y) -SQ(x) -sq(y)

aiB2(xi+y) -R with

R = Y, a,B2(xi) + E aiB2(yi) i i

_ a,B2(xi) + a,B2(y,)) +q0 E [n,a,,mia1] ® 1.

Here qo = q(q - 1)/2 and x = Y. , ni a y = X1 m1 aj, that is x,(1) = n,(1) and yi(1) = m, 1 with ni, mj E Z. Compare the commutator rule in Proposition 12.1.3. Since by Theorem 1.4.8 ai(B2(x,)+B2(y,)) = a,B2(xi+y,)+gon,miy(a,) ® 1 we thus get

O.,(x, y) = qol Y_ nim1[ ai, aj ] + nim;y(a1))® 1 i>j

=PAr(JAX 0 JAY) 392 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA where r is the retraction (for the basis of A ® Z/2 given by the elements a, (9 1). We can thus define the isomorphism

X: G,(q, A) = ?r2(7L/q, M(A,2)) by (x, b) - sq(x) + 0(b). Next we consider the case when A is any abelian group. Then A (& 7L/q is a bounded abelian group and hence a direct sum A ®71/q = 0, (/a,)a; of cyclic groups; compare for example Fuchs [I]. We now choose for theprojectionp: A -3, A ® 7L/q, q(a) = a ® 1,a map p: M(A, 2) - M(A (9 Z/q, 2). This map yields the following pull-back diagram of abelian groups

Ext(7L/q, F(A)) N ir2(7L/q, M(A,2)) -- Hom(7L/q, A) lr(P)' I6. 1P. Ext(7L/q,F(A (9 7L/q)) N ir2(7L/q, M(A (9 71/q, 2)) - Hom(7L/q, A (9 /L/q)

Here F(p) *is an isomorphism. Hence a splitting function sq for A ® 7L/q yields also a splitting function sq for A. This splitting, however, depends on the choice of the basis in A ® /L/q above and hence we cannot use Proposi- tion 12.1.9 for the naturality of sq. For the naturality of an isomorphism

X = X(qrG,(q, A) = ir2(7L/q, M(A,2)) itis enough to consider powers q = 2' of 2. Using the basis of A ® 7L/q above we obtain a retraction rq and a splitting function sq together with an isomorphism Xq: G,y(q',A) = ir,(7L/q',M(A,2)),q' 5q, which is natural in 7L/q' for q' < q, q' = 2". We set r = r2 and we define X(q) inductively as follows. For q = 2 we set X(2) = X2 Now assume X(q') is defined for q' = 2'> 2. Then we obtain X(q) for q = 2'+' by the composition

X(q) = Xq X: G,(q, A) = Go(q, A) = ?r2(7L/q, M(A,2)).

Here is the isomorphism given by rq - r as in the proof of Lemma 12.1.5. Hence X is natural in l/q and X(q)is natural in 7L/q' for q' 5 q. This completes the proof of Theorem 12.1.6.

12.2 Homotopy groups with cyclic coefficients

We compute the homotopy groups X) as functors in 7L/q. For this we need the following modification of the group G(q, A) in Section 12.1. 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 393

(12.2.1) DefinitionLet rl: A ---> B be a quadratic function which induces the homomorphism 17': T(A) -p B. Then the group G(q,'r) is defined by the product set G(q,,7) = Hom(7L/q, A) x Ext(7L/q, B). The group law (a, b) + (a', b') = (a + a', b + b' + 17 °A,(a®a')) is given by the cocycle A, in Definition 12.1.4. Since A, is natural in Z/q we obtain in this way the functor G(-,,i):fCyc°P-->Gr which carries 71/q to G(q, 77). Moreover we have a central extension

Ext(7L/q, B) N G(7L/q, rl) -µ-4 Hom(Z/q, A) of fCyc-groups as in(12.1.2). Clearly the universal quadratic function yA: A - I'(A) yields the group G(q, A) = G(q, yA) in Definition 12.1.4.

(12.2.2) TheoremFor a space X in Top* let 71= rte : rrn X -' vr + , X be the quadratic map induced by the Hopf map rt,,, n > 2. Then one has an isomorphism of groups X) a G(q, 17) which is natural in Z/q EfCyc. Moreover this is an isomorphism of central extensions compatible with A and A.

For the proof of Theorem 12.2.2 we need the following notation on central push-outs.

(12.2.3) DefinitionThe centre C of a group G is the subgroup of all elements g e G which commute with all other elements in G, that is for all x e G. A homomorphism a: A -* G is central if A is abelian and if aA lies in the centre of G. A commutative diagram

Pl c-push 1R B -- E of groups is a central push out if a and a are central and if for any pair of homomorphisms f:G-9L, g:B-+L, fa=gf3, 394 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA with g central, there is a unique homomorphism gyp: E --> G with cp# = f and cpa =g. We obtain E = (G X B)/ - from the product group G X B by the equivalence relation (x a(a), y) - (x, vi(a) y) with x e G, y e B, a E A. For example the diagram

Ext(7L/q,1'(A))N G(q, A) - Hom(Z/q, A) (12.2.4) 1`17 " ) . 1 II Ext(7L/q, B) N G(q, -q) - µ-> Hom(Z/q, A) is a central push-out diagram.

Proof of Theorem 12.2.2Let f: Y -* X be the (n - 1)-connected cover of X and for A = 7T,,X let g: M(A, n) -> Y be a map which induces an isomorphism Then we have the isomorphism

f*: X) and the push-out diagram (B =

Ext(7L/q, F, (A)) >-> ;r (7L/q, M(A, n)) -p Hom(l/q, A)

11. 1g II Ext(7L/q, B) ir,,(Z/q,Y) Hom(7L/q, A)

This yields, via (12.2.4) and Theorem 12.1.6, the isomorphism in the proposition; see Theorem 12.1.6(*) for n >_ 3. Compare also the proof of Theorem 1.6.11 which, however, is only available for n >_ 3.

Theorem 12.2.2 motivates the definition of the following algebraic category.

(12.2.5) DefinitionLet C be a full subcategory of JCyc. We define the C-enriched category FAb(C) of quadratic functions as follows. Objects are quadratic functions which are also the objects in FAb; see (7.1.1) For quadratic functions rl: A - B, ii': A' -> B' a morphism

('Po,(FI,F): rt-->71' in FAb(C) is given by a morphism Gpo, rp, ): n - rl' in TAb and by homomorphisms

F: G(7L/q, 17) --> G(7L/q, 77'),7/q E C, 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 395 for which the following diagram commutes and is natural in 71/q E C.

Ext 7L/q, B) N G(q, 71) - Hom(Z/q, A)

I F I(vo). Ext(71/q, B') - G(q, r1') - Hom(7L/q, A') Let SFAb(C) = FAb (C), n >- 3, be the full subcategory of stable quadratic functions in FAb(C) = rAb2(C). We obtain for n > 2 the functor (12.2.6) G = G,,: Top* , The functor G carries a space X to the quadratic function W = Tr,,(X) -' 7TR+1(X). Moreover G carriesa map f: X - Y totheproper morphism (7r .f, if, F): ijx - -qr given by with 7L/q E C. Here the isomorphisms are obtained by Theorem 12.2.2. Theorem 12.2.2 shows that G is a well-defined functor. The enriched category is part of a linear extension of categories

(12.2.7) N N I'Ab (C) -L I'Ab,,. Here ¢ is the forgetful functor which is the identity on objects and which carries the morphism ((po, Cpl, F) to gyp,). The natural system N is the bimodule on TAb given by N(q, j') = Natc(Hom(-, A), Ext(-, B')) where Hom(-, A), Ext(-, B'): C°P -* Gr are abelian C-groups and where Nat denotes the abelian group of natural transformations. For an element x E N(77, 77') with x: Hom(Z/q, A) -* Ext(71/q, B'),71/q E C, we obtain the action + of N by ((po, gPl,F) +x= (P, F+0xµ). Since G(q, r7) is part of a central extension we see that this is a well-defined action. We need the following natural transformation g between natural systems on TAb,,.

(12.2.8) g: E(q, -q')=Ext(A,B')- N(r7,77')=Natc(Hom(-,A),Ext(-,B')). 396 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

The map g carries x e Ext(A, B') to g(x) = y e N(-q, -q') where

y: Hom(Z/q, A) -> Ext(7L/q, B'), 7L/q E C, SI with y(ip)= cp*(x) for p* = Ext((p, B').

Compare (1.6.2). The next result on g is crucial for applications below.

(12.2.9) Lemma LetA be a direct sum of cyclic groups in C. Then gin (12.2.8) above is an isomorphism.

This is a slight generalization of Lemma 1.6.3. We are now ready to compare the linear extension for types' in (7.1.8) with the linear extension (12.2.7).

(12.2.10) TheoremLet n >_ 2 and let C be a full subcategory of fCyc. Then one has a map between linear extensions

+ E types;,k"' rAb

19+ 1G II N » 0 rAb

Here g is the transformation in (12.2.8).

The functor G in the theorem is given by the functor G in (12.2.6).

Proof of Theorem 12.2.10 We have to show that the functor G is equivariant with respect to g, that is G(f+a)=G(f)+g(a) for 0 E Ext(A, B') and f e [K67, n), K(77', n)]. For this we use the equivalence of categories

types;, = I'M" in Theorem 7.2.7. In fact, assume

f = E [ M(A, 2), M(A', 2)l = [ K(y, 2), K(y', 2)], where y, y' are the universal quadratic maps. Then we have for a e Ext(A, [A') G(irp+a)=G(ip)+g(a) 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 397

This follows since for i E G(q, A) _ [IP9, M(A,2)] we have the linear distributivity law in M2 (;p+a)1=;Px+x*(a) _;pz+Ag(a)p(z) where x = px = HZ x. Compare the proof of Theorem 1.6. 7.

We immediately derive from Lemma 12.2.9 and Theorem 12.2.10 the following results:

(12.2.11) TheoremThe functor G: types' -4 FAb (C) is full and faithful on the subcategory of all K(rl, n), n: A - B, for which A is a direct sum of a free abelian group and of cyclic groups in C.

(12.2.12) CorollaryLet Tl: A ---, B be a quadratic map where A is a direct sum of cyclic groups. Then the group of homotopy equivalences ((K(r),2)) is isomorphic to the group of automorphisms of 'q in the category FAb(JCyc). If 77 = yA: A --> FA we have C(K(yA,2)) = LE(M(A,2)).

12.2A Appendix: Theories of cogroups and generalized homotopy groups

The functor (12.2A.1) G,,: Top* - in (12.2.6) is a special case of a general concept of homotopy groups; see Theorem 12.2A.11 below. To see this we introduce the following general notation on theories. Recall that a contravariant functor F is the same as a functor F: COP --> K where COP is the opposite category of C.

(12.2A.2) Definition A theory, T, is a small category with a zero object * and with finite sums denoted by X V Y. Using * we have zero morphisms 0: X -p * -> Y for all objects X, Y in T. We say that T is a single-sorted theory generated by X E=- ObT if all objects of T are finite sums of X. That is, any object Y of T is of the form

Y= V XewithXe = X fore E E e e E where E is a finite set. This is the zero object if E is empty. Single-sorted theories were introduced and studied by Lawvere [FS]. We need theories which are not single sorted; they are also considered in Barr and Wells [TT]. 398 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

(12.2A.3) DefinitionLet T be a theory. A cogroup in T is an object X endowed with morphisms A: X - X V X, v: X --, X for which the following diagrams commute where 1 = lX. The morphism µ is the comultiplication and v is the coinverse. x

XIAN"' (1) X

AL1 11V µ (2)

XvX-'XvXVXµV1 X

(3) X4 XVX -X (1, V) (v.1)

The cogroup (X, µ, v) is commutative or abelian if the interchange map T: with Tit = i2 and Tie = i1, satisfies Tp. = µ in T. We say that T is a theory of cogroups if each object in T is a cogroup and if the cogroup structure of a sum X V Y is given by the composition XvY µv-- XvXvYvY=(XvY)v(XvY).

(12.2A.4) DefinitionLet T be a theory and let Set* be the category of pointed sets. A model G for the theory T is a functor

G:T°P-*Set* (1) which carries the zero object * in T to the zero object (0) in Set* and which carries sums in T to products in Set*; that is,

G(X v Y) = G(X) x G(Y) (2) is the product of sets where the inclusionsi,: X -* X V Y, i2: Y -* X V Y induce the projections p, = G(i,), P2 = G(i2) of the product set p,(x, y) =x, p2(x, y) = y for x E G(X), y E G(Y). Let modeKT) be the category of such models for T. Morphisms between models are the natural transformations of the corresponding functors. One readily checks the 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 399

(12.2A.5) LemmaLet T be a theory of cogroups and let G be a model for T. Then G carries objects of T to groups. That is, for any object X in T the set G(X) has the structure of a group which we write additwely. For x, y E G(X) addition is given by x + y = p.*(x, y) where we use (2) above. The negative of x is -x = v*(x) and the neutral element is the base point * of G(X).

For example for each object Y in a theory T one gets the model Dy=T(-,Y):T°P-Set* with ito (X)=T(X,Y), XEOb(T).

The base point of the morphism set T(X,Y) is the zero map 0. The universal properties of sums show that 0ysatisfies Definition 12.2A.4(2). We call 0y the model presented by Y. For any model G E modeKT) we have the Yoneda lemma

model(T)(Dy,G) = G(Y), (* ) that is, the natural transformations F: Dy --> G are in 1-1 correspondence with the elements f c= G(Y). The correspondence carries F to f = F(ly) with 1y E Dy = T(Y, Y). Moreover the Yoneda lemma shows that one has a faithful Yoneda functor

D : T -> model(T), (* * ) which carries an object Y to 0 y = T(-, Y) and which carries a morphism g: Y -> Y' in T to the induced natural transformation g * : T(-, Y) -+ T(-, T') given by g * f = g o f for g E T(X, Y). Now let T be a theory of cogroups. Then T(X, Y) is a group. For a morphism the induced map g*:T(X,Y)-->T(X,Y'), g*h=gh, (1) is a homomorphism between groups. For f: X' - X the induced map f*:T(X,Y)--,T(X',Y), f*h=hf, (2) need not be a homomorphism. Here f * is a homomorphism between groups for all Y if and only if the diagram

f X' X

(3) X'vX' commutes. In this case we say that f is a morphism of cogroups in T or f is linear. In a similar way we see that for any model G the induced map 400 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA f G(X) --> G(X'), in general, is not a homomorphism between groups; but f *is a homomorphism if f is linear. Moreover we observe that for a morphism z: G - G' between models the induced map rx: G(X) -> G'(X) is always a homomorphism between groups. We now are ready to define `generalized homotopy groups'.

(12.2A.6) DefinitionLet £ be a class of spaces in Top* and let 1.EI be the full subcategory of the homotopy category Top*/= consisting of all finite one-points unions X, V ... V X, with X; E .for i = 1,... , r. Moreover let F,".be the class of n-fold suspensionsy3E" = {I"X; X E}. Then [E"] is the homotopy category of all one-point unions "(X,V...VX,)="X,V...VI"X,. (1) The one-point union is the sum in the category [I" 1] for n 2t 0. Hence the category [I" .£] is a theory, in fact, a theory of cogroups for n >- 1 since the suspensions "X are cogroups by the classical comultiplication V X - y," X V In X. Recall that [ X, Y ] is the set of homotopy classes of base-point preserving maps X - Y. The generalized homotopy groups or 3-homotopy groups are the functors (n > 0)

ir,,: Top*/= - model[ In X (2) which carry a space X to the model Mx = 1r,,r(X) with MX=[-,X]:[I".T]-Set* with StMX(yn(X, V... VX,))=[1"(X, V ... VX,),X]. (3) Clearly MX carries a sum to a product. A map f: X - Y in Top*/= induces in the obvious way a natural transformation f*: MX - My. (12.2A.7) Example Consider the class . = (S°) which consists only of the 0-sphere S°. Then we have canonical isomorphisms of categories model[{S°}] = Set*, model[I(S°)] = Gr, model[F,"(S°)] = Ab, n> 2, where Gr and Ab are the categories of groups and abelian groups respectively. Moreover the functorRs°)can be identified with the classical homotopy groups 7'r0(s°) = a°: Top*/= - Set*,

,,Is') = ir, : Top*/= -* Gr,

7T,.'}=s° rr":Top*/-- Ab,n>_2. 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 401

In this sense X-homotopy groups above generalize the well-known homotopy groups ,7r,,with ar"(X) _ [E" S°, X1.

(12.2A.8) Example For .)` _ (S', S2, ...) consisting of all spheres S", n >_ 1, the category model(I X] is the category of ir-modules used by Stover [KS], Dwyer and Kan [irA]. Using the functor ir,;r Stover describes a generalized Van Kampen theorem in terms of a spectral sequence; see also Artin and Mazur [VK] and Dreckmann [DH].

(12.2A.9) Example For I = {SQ, St, ...1 consisting of all rational spheres Sa, n z 1, the category modeK13() is equivalent to the category of graded rational Lie algebras concentrated in degree >_ 1. This is the basic example for rational homotopy theory. In Baues [CC] we also consider modeKY. I) for I = (S',... SR',...) consisting of all localized spheres SR, n > 1, where R c Q is a subring with 2, 3 E R.

Let C be a full subcategory of Cyc and let

(12.2A.10) 13 = $ (C) _ {S', S2, Pq; l/q E C) be the class of spaces consisting of the 1-sphere, the 2-sphere, and the pseudo-projective planes P9 for which 71/q is an object in C. In this case we get the following result which shows that the functor G" in (12.2.6) can be identified with i, ,.

(12.2A.11) TheoremThere is a canonical full inclusion of categories (n >_ 2)

rAb"(C) c model[ I" - ' such that the diagram of homotopy functors

Top*'" 1 model[I"-''43] U it Top* G" rAb"(C) commutes.

Proof We observe by Theorem 12.2.11 that

G = G" : [ E"' $1 -+ (1) 402 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA is a full and faithful functor. Hence we obtain the functor

j: rAb"(C) --> model [X" '13] (2) which carries q to the model

M,,: [i"-''13] - set* M,,(Y) = [G(Y),71] = [Y, K(ii,n)] where y= yn-'(X, V VX,), X, E 13, and where [G(Y), q] is the set of morphisms G(Y) in FAb"(C). Since G in (1) carries sums to sums we see that Mn is a model and that the functor j in (2) is well defined. Now one can check that j isfull and faithful and that the diagram in Theorem 12.2A.11 commutes. In fact, for q: A --+ B the model M,, satisfies

M,,(I"-' S') =A,

M,,(l"-' S2) =B, M, (!`1P,) =G(q, ri). 11

(12.2A.12) Example Unsold [AP] studied the generalized homotopy groups n >_ 1, for . = {S2, S3, S4, CP2} where CP2 is the complex projective plane. This leads to the classification of homotopy types of (n - 1)-connected (n + 4)-dimensional CW-complexes with torsion-free homology, n > 3.

12.3 The functor t4

In Section 11.3 we computed F4(X) as an abelian group in terms of the quadratic function i = 71X: 1r, X - 1r3X induced by the Hopf map 712. Here we describe the functorial properties of r4(X). For this it suffices to consider X = K(r1, 2) where K(71,2) is the 1-connected 3-type given by 71. Let PCyc be the full subcategory of Ab consisting of cyclic groups 7L/q where q is a power of a prime. We consider the diagram of functors

typesZ -L FAb(PCyc) (12.3.1) r\Ab/4

Here G = G, is the functor in (12.2.8) which carries K(r1, 2) to 11 and I'4 is the functor in Whitehead's exact sequence. 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 403

(123.2) TheoremThere is a functor T4 in (12.3.1) together with a natural isomorphism 0: T4G = I.

We shall describe the functor f4 purely algebraically by defining T4(q) in terms of generators and relations. We have the short exact sequence

(12.3.3) 172267) -TT(A) for n: A -b B E Ob(TAb). The sequence is natural in K(q, 2) E types; see Section 11.3. We first describe the torsion functor FT in terms of generators and relations, then we choose generators in I'4K(7,2) which map via µ to generators in TT(A). This leads to the definition of T4 below.

(12.3.4) Definition We define the functor

PT: Ab -> Ab as follows. The group rT is generated by the symbols

z x E Hom(7L/2, A) [x,Y] ® ij E Hom(7L/n, A) with 7L/n E PCyc. Induced maps for (p: A -A' E Ab are defined by

)0 e2 w*([x,y](D f )=[0X,9]®

The relations in question for the generators in (*) are given by the following list (a)-(e): (a) [i,XJ®=0; (b) [x,X]®e,,, = -[9,i]® Let x, : i/k - i/n be the canonical generator in Hom(Z/k, i/n). Then: t (C)[X,(Xn n`)*y] ® Snm..nm = [(Xnm)*X, 9] ® 5n,n; (d) (x+y)®e2=X®f2 +y0 [S2+[Z,y]®52,2; (e) [x, y] is linear in z and y.

(1235) Lemma There is a natural isomorphism

FT(A) = FT(A). 404 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

Proof The isomorphism carries z (& 62to y(z) E I'(A *1/2) and carries [X, y] ®4n.n to Tn(X, y) EA * A. Here we use the generators of FT(A) in the proof of Theorem 6.2.7.

We now use the isomorphism G(n,,q) = [I Pn, K(,l, 2)] in Theorem 12.2.2 which is natural in Z/n E PCyc. Moreover we use the diagram

Ext(71/n, B)°G(n,,]) Hom(71/n, A) (12.3.6)

B A where we identify Ext(l/n, B) = i/n (D B and Hom(Z/n, A) = i/n * A CA. The generators in Definition 12.3.4 correspond to the following composites which we consider as elements in F4 K(,t, 2),

S4->y62 P2-->K('h,2)3 (12.3.7) 4 [x,y] 3 [X, y] ® n n: S -- 2 r APn - OK(f,2). Here [x, y] is the Whitehead product for x, y E G(n, rl). The elements 21 Sn.n are the maps described in Section 11.5. One readily checks that µ in (12.3.3) satisfies

µ([x,y] ® n.n) _ [i,9] ®Sn.n where i = µ(x) is given by µ in (12.3.6). Now the exact sequence (12.3.3) and Lemma 12.3.5 show:

(12.3.8) Lemma The abelian group I'4 K(,l, 2) is generated by elements A( p) with p c- I'2(71) and the elements x ®f2, [x, y] ®Sn.,above. We want to describe the relations of the generators in Lemma 12.3.8. For this recall that (12.3.9) r2(,,)=(B®7L/2®B®A)/- is obtained by the equivalence relation in Definition 11.3.3. For a E A, b r= B we thus have the elements {b ®1}, {b (9 a} E I'2(,1) (1) represented by b® 1 E B ®1 /2 and b® a E B® A respectively. For x, y E G(n,,t) we get i = µ(x), y = µ(y) EA by (12.3.6) and the quadratic function ,1 yields [z, Y],, = ,t(z, y) -,q(i) - 77(y) E B (3) 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 405 as in Definition 11.3.3(5). We now obtain the abelian group r4(q) by describing generators and relations.

(12.3.10) Definition We define the algebraic functor I'4 in (12.3.1) together with a natural short exact sequence

F2(71)) " ) F4(rl) FT(A) (*) for q: A --> B. The group T4(77) is generated by the symbols x®;t2, xEG(2,z) [x,y]®Sn,n,x,yEG(n,rl),Z/nePCyc (**) A( p),p E F q).

Moreover 0 in (*) carries p to A(p) and µ in (*) is given by (AA( p) = 0 and (12.3.7). A morphism

x = (po, (p1, F): n -* n' E I'Ab(PCyc) as in Definition 12.2.5 induces the homomorphism

F4(X) = X* : r4(77) - F40') defined on generators by

X* (X (&2) _ (Fx)

X*([x,y]®fn,n)=[Fx,Fy]®Snn (***) x*o(p)=or2((Oo,co,)(p).

The relations in question for f4(,1) are given by the following list (a)-(f):

(a) [x, x] ®n,n = 0; (b) [x, y] [y, x] ® -t (c) [x,(X, m)*y] ®Snm.nm = [( Xnm) x, y] ®n. n' For this compare Definition 12.3.4(c) and the functorial properties of G(n, r)) in i/n in Section 12.2. (d) (x+y)(& 62=xC2+y®1;2+[x,y]®f2,2+O(Pi) wherep, _ ([x,],r ®y} E r2(77); (e) IX where P2 = 0 if n is odd and P2 = (n/2)([i, y],, (&y} E F2(i) if n is even; 406 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

(f) A in (*) is an injective homomorphism and for b E B the element Ab given by (12.3.6) satisfies (Ab) A(b ® 1}, [Ab,x]® ,,,,,=A{b®z}. This completes the definition of the functor r4 in (12.3.1) and Theorem 12.3.2. The exactness of (*) is a consequence of Lemma 12.3.5 since killing the elements A( p) in the relations above yields exactly the relations in Definition 12.3.4.

Proof of Theorem 12.3.2 We define the isomorphism

6: I7467) = f4K(71,2) on generators in the obvious way by BA( p) = A( p), see (12.3.3), 0(x ® 2) _ X and 6[x, y] ®& _ [x, y] ®see (12.3.7). The formulas in Sec- tions 11.5 and A. 10 show that 0 is a well-defined homomorphism compatible with A and µ; this shows that 0 is an isomorphism since Definition 12.3.10(*) is exact. In fact, 0 is well defined since we use Lemma 11.5.15 for (a) and (b) and we use Lemma 11.5.24(b) for (c). Moreover (d) is obtained by Theorem A.10.2(b) since [[x, y1, y]T212 62,2 corresponds to p,; see Lemma 11.5.27. Next we get(e) by (11.5.10)(6), Lemma 11.5.27 sinceP2corresponds to -[y, [x, z]]X(A A 1)6,,,,, = [[x, z], y]E(A A Finallyq* 12 = 73in (11.5.25) yields the first equation in (f) and the diagram in the proof of Lemma 11.5.27 yields the second equation in (f ).

12.4 The bifunctor F3

In Section 11.3 we computed 173(H, X) as an abelian group in terms of the quadratic function 71 = 71X: '2 X --> ir3 X induced by the Hopf map i 2. Here we describe the functorial properties of r3(H, X) provided the homology H2X is finitely generated. We proceed similarly as in Section 12.3. Again it suffices to consider X = K(q, 2) where K('q, 2) is the 1-connected 3-type given by 71. Recall that we have the equivalence of categories

G: M3 = G where M3 is the homotopy category of Moore spaces in degree 3 and where G is the algebraic category in Theorem 1.6.7. We now consider the diagram of functors (M3)°a cxc x ti GOP x rAb(PCyc) 2 (12.4.1) ,l Ab 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 407 where we use the functor G in (12.3.1) which carries K61,2) to rl. The functor t3 carries (M(H, 3), K(rl, 2)) to the abelian group I'3(H, K(rl, 2)) in Section 2.2.

(12.4.2) TheoremThere is a functor P3 in (12.4.1) together with a natural transformation 0: r3(G x G) - I', such that 0: F3(H,rl)->F3(H,K(r1,2)) is an isomorphism if the group A, given by rl: A - B, is finitely generated.

We shall describe the bifunctor T3 purely algebraically by defining P3(H, q) in terms of generators and relations. For this we use the short exact sequence

(12.4.3) Ext(H, I'46l))>-°-a F3(H, K(71,2)) - Hom(H,1'(A)) given by the universal coefficient sequence and by Theorem 12.3.2. The sequence is natural in K(rl, 2) E types'' and H E G = M3. We first describe the bifunctor (H, A) - Hom(H, F(A)) in terms of generators and relations, then we choose generators in r3(H, K(rl, 2)) which map via µ to the generators in Hom(H, I'(A)). This leads to the definition of F3 below.

(12.4.4) Definition We define the bifunctor

P: Ab°P X Ab -4 Ab as follows. Let PCyC° be the full subcategory of Ab consisting of 71= 7L/0 and cyclic groups 71/q where q is a power of a prime. For 7L/n, 7L/m in PCyc° we have the homomorphisms T:71/n®7L/m=7L/m(9 71/n I'(71/n) -H71/n ® 71/n P I'(71/n) where T is the interchange map and where H and P = [1, 1] are defined as in Section 1.2. The group F(H, A) is generated by the symbols X®yn®a (*) [i,$1 ® ln.m ®b where i e Hom(7L/n, A) and a E Hom(H, I'(7L/n)), y E Hom(7L/m, A) and b E Hom(H,Z/n ® 7L/m) with 7L/n,7L/m E PCyC°. Induced maps are defined by 41*w*(x®yn®a)=((Pi)®y,,®(ut/r) (**) 0*w*([X,y]®*In.m®b)=[ci,wy]®71n.m0(b) 408 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

The relations in question are:

(a) ®b=.®yn ®(Pb); (b) [.,y]®11n,m ®b= [9,x]®91m,n ®(Tb); (c) for the generator x = xk E Hom(7L/k,7L/n) we have (x*.)® yk®a=X® yn®r(x)*a [x*z,Y]®71k,m®b=[i,9]®%.m®(x®1)*b; (d) (.+y)®y,, ®a =.® yn ®a+y ® yn ®a+[i,y]®'nn (Ha);

(e) [1,9]® 71nm®b is linear in ., y; (f) i ® yn ® a and [X, y] ® 71n m ® b are linear in a and b respectively.

(12.4.5) Lemma There is a natural transformation 0:P(H,A)-+Hom(H,FA) which is an isomorphism if A is a finitely generated abelian group. Proof We define () by 0(.®yn®a)=r(.)a

0([i, 10 71"'M ®b) _ [X,Y]b where [., y] _ [1,1x. ®y) is defined as in (1.2.4). The proposition now is a very special case of 4.4. in Baues [QF]. 11

We now use G(n, r1) in (12.3.6) and Theorem 12.2.2 and we set G(0, r1) =A for n = 0. The generators in Definition 12.4.4 correspond to the following composites of maps which we consider as elements in 1'3(H, K(71,2)); see (2.2.3)(2), (12.4.6)

x®y, ®a:M(H,3) ° +M(r(7L/n),3) .Pn x K(i,2)3 'YP I --K( [x,y]®iln.m®b:M(H,3) b >M(7/n®71/m,3) APn, - ,1,2)3. Here [x, y] is the Whitehead product for x e G(n, q), y E G(m, r1) and 7L/n, i/m E PCyc°. The maps yn and '11n, m are the generators in the corresponding elementary homotopy groups, see (11.5.16). For this we identify in the canonical way the groups r(i/n) and 7L/n ® Z/m with the corresponding cyclic groups so that M(I'(i/n),3) =12 Pkwithk = (n2,2n), (1) M(7L/n ® 7L/m, 3) = 2 Pkwithk = (n, m). (2) 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 409

Using these identifications we set y,, = yk and 71,,m = 9,, m.Clearly for n = 0 the map yo: S3 _ S2 is the Hopf map. One readily checks that µ in (12.4.3) carries the elements (12.3.7) to the corresponding elements in Hom(H, I'A), that is µ(x®y,,®a)=O(z®y,,(9 a) (12.4.7) µ([x,y] ® 1n.m ON = o([x,y] ®11nm®b).

Here we set i = µ(x) as in (12.3.6), and we set µ(a) = a and µ(b) = 6 where we use µ in the universal coefficient sequence

(12.4.8)

Ext(H,7L/k(9 7L/2))° ;[M(H,3),V Pkj -Hom(H,7L/k) which is natural in 7L/k E PCyc since we can use B3 in Corollary 1.4.6. Using G in (12.4.1) we identify [M(H, 3), j,2Pk] = G(H,7L/k).

(12.4.9) Lemma The elements (12.4.6) and the elements 0( p) with p E Ext(H, f4(,t)) generate the abelian group F3(H, K(i, 2)) provided the group A is finitely generated with rt: A -> B.

This is a consequence of (12.4.7), Lemma 12.4.5 and (12.4.3). Below we describe explicitly a complete set of relations for the generators in the lemma. We need the following notation. Given a homomorphism f: H -+ 7L/k in Ab and an element u E U = I'4(q) let

(12.4.10) f *u =f *(1 0 u) E Ext(H, ['4(77)).

Here 10 u E 7L/k ® U = Ext(7L/k, U) is represented by u.

(12.4.11) Definition We define the algebraic bifunctor I'3in (12.4.1) together with a natural short exact sequence

Ext(H,I'4(rl))> *['3(H,i) -I'(H,A) (*) for ,t: A ---> B. The group NH, n) is generated by the symbols (7L/n,7L/m E PCyc) x®yn®a, xEG(n,q), aEG(H,1'(7L/n)) [x,y]®,7n,m®b, xEG(n,rl), yEG(m,q), bEG(H,7L/n®7L/m) 0(p),p E Ext(H, F4(1])).

(**) 410 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

Moreover A in (*) carries p to A(p) and µ in ( * ) is given by uA( p) = 0 and (12.4.7). A morphismip = ((p, ip): H' - H in G and a morphism X = (90, (p,, F): 71 -> r ' in FAb(PCyc) induce the homomorphism *X* r3(H,r)-I'3(H',71') defined on generators by ip*x(( p)) = A((p*r4(X)* (p)) rv*X*(x(9 yn®a)=(x*x)®yn®(;p*a) (***) w*X*([x,yl ®1n.m(9 b)=[X*x,x*y]®77n,m® (;p*b).

Here we set X * x = Fx for n > 0 and X* x = (poxfor n = 0 and x e G(n, q). The relations in question for P3(H, r1) are given by the following list (a)-(g):

(a) [x, x] ®rt,,,,, ®b =x ®yn ®(P* b) + A(b*p,)wherep, E ra(il)is given by

P, = 71(z) ® 1). Here we set S = 1 for n = 4 and 6, = 0 otherwise. (b) [x, yl ®qn,m ®b = [y, x] ®rlm,n ®(T* b) + A(b*P2) where p2 E r4(71)is given by P2 = -[y,x]®6n n+SnA{[x,9], ®1) for n = m > 0 and p2 = 0 otherwise. Here we use k as in (a). We point out that with the identification in (12.4.6)(2) the map T* is actually the identity. (c) For the generator X = Xk E Hom(7L/k,7L/n), n not equal to k, we have (X*x) ® Yk ®a =x® y, ®(r(X)*a), [x*x,y]® lk,m®b=[x,y]® 1n,m®((x®1)*b)+A(b*P3). Let 8,,,,n = 1 if n and m are powers of the same prime and m > n and let Sm =0 otherwise. The term p3 E r4(,1) in the second equation is given by P3 = 0 if m = 0. Moreover for m, n not equal to 0 let n S [x,(Xn)*y]®fn,n+e fork=0or(k,m)=m (k, n) n.m P3 =J -k t otherwise I. (k, n)sm,n[( X,, )*x,yl ®Sm,m

where 6 = A{[x,Y],, (& 1} E 17401) 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 411

if m = 2 and n = 2k where k is a power of 2; moreover e = 0 otherwise.

(d)

+[x,y] (H*a)+A((Ta)#P4). Here r: F(7L/n) -' F(7L/n) is the isomorphism 1/2 if n is odd and is the identity otherwise. Moreover p4 E I'4(r7) is given by p4 = 0 for n = 0 and p4 = -[x, y] ® + e for n not equal to 0 where e = (n/2)A{[z, fl,, (9y} if n is a power of 2 and e = 0 otherwise.

(e)

[x +y, z] ®11n.m ®b= [x, z] ®7lr.m ®b+[y,z] ®fn.m 0 b+A(b#P5). Here p5 E F4(i) is given by p5 = (n/2)A{[z, ij,7 ®y} if m = 0 and n a power of 2 or if m, n are powers of 2 with m > n. Moreover p5 = 0 otherwise. (f) x ® y ®a and [x, y] ®,q,, m ®b are linear in a and b respectively. (g) A in (*) is an injective homomorphism and A in (12.3.6) satisfies for x' EB,n>0, (Ax') 0®a=0

Ax', y] 0 '1R.m ®b- A(b#P5) where p5 E F4(q) is given by P5 = - A[x' ®y} if m > n or m = 0; and p5 = 0 otherwise. Finally A in (12.4.8) satisfies for a' E Ext(H,7L/n ® 1/2) x®y 0 (Aa') = A( p6)*(a') where p6: 1/n 0 71/2 - r4(rl) carries the generator to A{i7(i) 0 1). More- over for b' E Ext(H,71/n 0 71/m 0 71/2) we have [x,y] ®rl,,.m ® (Ab') = A( P7) * (b') wherep,: 71/n 0 Z/m 0 71/2 --> 174(rl)carriesthegeneratorto A{[X, yl,, ®1).

This completes the definition of the functor 1'3in (12.4.1) and Theorem 12.4.2. The exactness of (*) above is a consequence of the definition of r in Definition 12.4.4 since killing the elements A( p) in the relations above yields precisely the relations in Definition 12.4.4. 412 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

(12.4.12) Remark We have the exact sequence

Ext(H, F22(77)) --b P3(H, 17) --p FT,(H, A) -> 0 where we assume that A is finitely generated. Hence in this case we obtain by (12.4.11) also generators and relations for the bifunctor FT,r(H, A). For this we only need to kill all curly brackets in the relations of (12.4.11) above.

Proof of Theorem 12.4.2 We define the transformation

0: F3(H, i7) ---> F3(H, K(-q, 2)) on generators in the obvious way by OA( p) = 0( p); see (12.4.3), and (12.4.6). The formulas in Sections 11.5 and A.10 show that 0 is a well-defined homomorphism compatible with A and µ. If A is finitely generated this shows by Lemma 12.4.5 that 0 is an isomorphism since (12.4.11X *) is short exact. In fact, 0 is well defined since we use Lemma 11.5.19 for (a) and we use Lemma 11.5.20 and (11.5.103) for (b). Next we obtain (c) by Lemma 11.5.24; for the second equation this is a somewhat nasty computation. We derive (d) from Theorem A.10.2(b), (11.5.16X7), Lemma 11.5.24(e), Lemma 11.5.27. We derive (e) from (11.5.10X6) and Lemma 11.5.27. Finally we obtain (g) by the definition of y and

12.5 Algebraic models of 1-connected 5-dimensional homotopy types We introduce the algebraic category of A3-systems and we describe a detecting functor from the homotopy category of simply connected 5- dimensional CW-spaces to the category of A3-systems. Hence isomorphism types of Az-systems are in 1-1 correspondence with 1-connected 5-dimensional homotopy types. The definition of A3-systems here depends on the highly intricate functors F4 and 173 in Sections 12.3 and 12.4. In the following sections we describe special cases for which these functors are obtained more directly by algebraic constructions. Recall that quadratic functions q: A --> B, or equivalently homomorphisms F(A) -> B, are the objects of the category TAb; see (7.1.1). Moreover G is the category equivalent to the homotopy category of Moore spaces M', n z 3, in Section 1.6. The objects of G are abelian groups H E Ab. We shall describe various examples of algebraic categories K and functors

k: K -+ TAb (12.5.1) F4 : K - Ab I73:G°PxK--Ab 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 413 together with a short exact sequence

Ext(H, r4(,)) -L r3(H,17) - Hom(H, F(A)) which is natural in H E G and 71 E K. Here the objects r1 in K are also objects of TAb andk: Ob(K) C Ob(T(Ab) isan inclusion. Given such data (K, k, r4, r3, A, µ) we introduce the following algebraic objects.

(12.5.2) Definition An AZ-system over K

S = (H21 H41 H51 ir3, 64,71, 65, /3) (1) is a tuple consisting of abelian groups H2, H4, H5, 7T3 and elements

b4 E Hom(H4 , r(H2 ))

i1 E Hom(r(H2), Tr3)with 7E=-0b(K) (2) b5 E Hom(H5, r4(r1))

/3 (=-r3(H4, 7,)/0(65) * Ext(H4, H5).

The elements satisfy the following conditions (3) and (4). The sequence

-77 H4b4FH2 -0 7r3 (3) is exact and

µ(f3) = b4 (4) where µ is the operator given by µ: r3(H4, 71) -+ Hom(H4, r(H2)) in (12.5.1) above. A morphism

(P2, P4 + V5, V , Vr) : S --> S' (5) between A2-systems is a tuple of homomorphisms in Ab

cp,: H,-->H; (i=2,4,5) 9,: 7r3 -4 IT 3 9r: r4(71) -+74(11') 414 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA such that the following properties are satisfied. The diagram

b4 H4 , I'(H2) n'1r3

Iw< Ir(v,) IV, (6) H4 ----> r(HZ) - 1r3 b4 71 commutes and also

(7)

commutes. Moreover, there is a morphism x: ,q - 71'in K with k( x) = (cp2, see (12.5.1), and there is a morphism ('4): H4 -> H4 in G such that

(Pr=r4(x): f4('q)->I'4(0 (8) is induced by x and such that

(9) in I'3(H4,'q')/o(b5)* Ext(H4, H5). Here X. and ((p4, ip4)* are the homomorphisms induced by the functorI'3. There is an obvious composition of morphisms so that the category of A;-systems over K is well defined. An AZ-system S as above is free if H5 is free abelian and S is injective if b5: H5 --* I74('q) is injective. Let A2'-Z Systems(K) resp. A23-systems(K) be the full subcategories of free, resp. injective, A2'-systems. We have the canonical forgetful functor

0: A32-Systems(K) -*A3-systems(K)2 (10) which replaces b5 by the inclusion b5(H5) c F4(61). One readily checks that 4. is full and representative.

(12.53) Definition We associate with an Az-system S the exact I'-sequence b, H5 I'4(77)- 1r4 '--b H4bsa ['(H2) - + 1r3 -* H3 -+ 0.

Here H. = cok(7I) is the cokernel of 71 and the extension

cok(b5) H 1r4 ker(b4) 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 415 is obtained by the element 8 in Definition 12.5.2(1), that is, the group 1r4 = Tr( ft) is given by the extension element

Pt E Ext(ker(b4),cok(b5)) defined by Qt= (j,J)*(0) Here j: ker(b4) c H4 is the inclusion and (j, j): ker(b4) - H4 is a morphism in G which induces (j, j)* via the functor r3; compare (2.6.7). Let types2(K) be the full subcategory of the category of 1-connected 3-types consisting of all K(i7, 2) with q7 E K. We say that the structure (12.5.1) is good if there is a full functor

(12.5.4) r: types' (K) - K which carries K(q, 2) to 17 such that the composite functors

7 typesZ(K) -----* Kr'-> Ab b-- (M3)°P x types' (K) -G-- GOP x K Ab are naturally isomorphic to the homotopy functors F4 and r3 respectively, that is (a) f4K(ii,2) =1'4(7) and (b) r3(H, 2)) = r3(H, rl) and these isomorphisms are compatible with A and µ in (12.5.1) and Definition 2.2.3(3). Moreover the composite functor types;(K) K -L fAb is naturally isomorphic to the functor k2 which carries K(77,2) to 17; see Proposition (7.1.3).

(12.5.5) ExampleLet K be the full subcategory K c rAb(PCyc) consisting of all objects 17: A -- B for which A is finitely generated. Then we obtain by G in (12.3.1) the equivalence of categories

r: typesz(K) -:4 K 416 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA where types2(K) is the homotopy category of all K(-q, 2) with 71 E K. More- over using the functors F4 and r3 in Sections 12.3 and 12.4 we obtain a good structure as above. For a category K as in (12.5.1) or (12.5.4) let spaces'(K) be the full homotopy category of 1-connected 5-dimensional CW-spaces X for which the quadratic function rlx=(12)*: '7r2X=H2X-> ir3X is an object in K. Let typesz(K) be the corresponding category of 1- connected 4-types. We have the Postnikov functor

(12.5.6) P: spacesz(K) types2(K) which carries X to its 4-type. The next result can be applied to the example in (12.5.5) above.

(12.5.7) Classification theoremGiven a category K and a good structure as in (12.5.4) and (12.5.1) there are detecting functors

A': spaces2(K) -->A;-Systems(K) A': typesz(K) -Az-systems(K).

Moreover there is a natural isomorphism 4)A'(X) =,1'P(X) for the forgetful functor ¢ in Definition 12.5.2(10) and the Postnikov functor P in (12.5.6). Proof Let C= types'' (K) be the category in the classification theorem (3.4.4). The bype functor F on C defined in (3.4.3)(3), n = 4, leads via the good structure on K to the following bype functor F' on K. Let F': Ab°P X K - Ab be defined by the pull-back diagram, H E Ab, -1 E K,

Ext(H, I74('q)) N F3(H, r,) - Hom(H, F(A))

(I U U Ext(H,F,(rt)) NF'(H,71)- Hom(H,Fo-q) where F,(rt) = r4(,,) and FF(rl) = ker(rt: r(A) - B). Induced maps for F' are 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 417 defined via the induced maps for the functor F3 in (12.5.1) and (12.5.4). It is clear that we have detecting functors Bypes(C, F) - Bypes(K, F') -T-. A 3 -Systems(K). The detecting functor T is induced by the full functor T in (12.5.4). Moreover the detecting functor T' is the `forgetful' functor. The functors T and T' are essentially the identity on objects and by definition they are both full functors. Now the classification theorem 3.4.4 yields the proposition of the theorem.

Using the isomorphisms (12.5.4)(a), (b) as identification we define the detecting functor A' in Theorem 12.5.7 by (12.5.8)A'(X)=(H2X,H4X,H5X,Ir3X,b4X,,qx,b5X,/3(X)) where b4 X, b5 X, -1x are part of Whitehead's exact F-sequence with r3(X) _ F(H2 X) and where /3(X) = /34(X) is the boundary invariant of X. Similarly we define the detecting functor A'. The detecting functor A' in Theorem 12.5.7 shows that for each free AZ-system S over K there is a unique 1-connected 5-dimensional homotopy type X = Xs with A'(X) a S. Then the F-sequence for S in Definition 12.5.3 is the top row in the following commutative diagram

H5 -* ir4 - H4 - I'(H2) -'ir3- H3

II II 1 II II II II H 5 X - F4 X - 7r 4 X - H4 X -'F3 X --> or3 X -* H3 X The bottom row is Whitehead's exact f-sequence of X. The diagram describes a weak natural isomorphism of exact sequences; see (3.2.5)(5). If we apply Theorem 12.5.7 to Example 12.5.5 we get the following crucial result.

(125.9) Classification theoremLet K c FAb(PCyc) be the full subcategory of all q: A -a B for which A is finitely generated and let an AZ-system over K be defined by the functors T4 and I3 in Sections 12.3 and 12.4. Then there is a detecting functor from the full homotopy category of all simply connected 5-dimensional CW-spaces X with H2 X finitely generated to the category A 2-System(K).

Hence this result yields in particular algebraic models of all homotopy types of finite polyhedra which are simply connected and of dimension 5 5. In the following sections we describe applications of Theorem 12.5.7 for which the functors I74, F3 are less complicated. 418 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 12.6 The case ir3X = 0

We consider Az systems which correspond to simply connected 5-dimensional homotopy types X with ir3 X = 0. They turn out to be the same as certain bypes used in Theorem (6.4.1). Let K = Ab be the category of abelian groups. Then we obtain the functors k: K - FAb (12.6.1) r4=rT: K -->Ab r3: GOP x K - Ab°P x K - Ab as follows. Let k be the inclusion which carries an abelian group A to the quadratic function rl: A -> B for which B = 0 is trivial. Moreover r4 = FT is the I'-torsion functor and r3 is given by the projection G - Ab and by the torsion bifunctor FT,,. See Definition 6.2.11 where one also finds the short exact (0, µ)-sequence for FT#. Hence the functors in (12.6.1) form a structure as in (12.5.1) and the AZ-systems over K are defined by (12.6.1). Using the equivalence T: types' (K) -= K which carries K(rl, 2) = K(A, 2) to A we see that this structure is good in the sense of (12.5.4); compare Section 6.3. Hence we can apply Theorem 12.5.7. For this we have the identification

Az-Systems(K) = Bypes(Ab, FT,) of categories so that the following corollary of Theorem 12.5.7 is also a consequence of Theorem 6.4.1.

(12.6.2) TheoremLet AZ-systems be defined by the structure on K = Ab in (12.6.1) above. Then there is a detecting functor from the homotopy category of simply connected 5-dimensional CW-spaces X with ir3X = 0 to the category A Z-Systems(K).

12.7 The case H2 X uniquely 2-divisible

We here consider simply connected 5-dimensional homotopy types X for which HZ X is uniquely 2-divisible. Our method is not the same as the approach by `tame homotopy theory', see Dwyer [TH], Anick [HA], and Hess [MT]. The results in this section yield new insights into why tame homotopy theory works; moreover comparison of the results here with the case in Theorem 12.5.9 sheds light on the kind of complexity appearing outside the tame range. 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 419

An abelian group A is uniquely 2-divisible if multiplication by 2 is an isomorphism 2: A -A; the inverse of this isomorphism is denoted by 1/2. Let 7[1/2] be the smallest subring of the rationals 0 containing 1/2 E Q. Then A is uniquely 2-divisible if and only if A is a Z[1/2]-module or equivalently if A *71/2 =A ® Z/2 = 0. Let Ab[1/2] be the full subcategory of Ab consisting of uniquely 2-divisible groups. Moreover let M2[1/2] be the full homotopy category of Moore spaces M(A, 2) with A E Ab[1/2].

(12.7.1) PropositionThere is a functor s: Ab[1/2] _ M2[1/2] which is a splitting of the homology functor H2.

Proof Let rp E 2 Hom(A, B), say cp = 2 tli. We construct an element s((p) E [M(A,2), M(B,2)] = G (1) as follows. We choose homotopy equivalences M(A, 2) _ XX, M(B, 2) = Y.Y for appropriate X = MA and Y = MB. Then the map - I I X : I X - F X induces the inverse, - a, in the group [I X,1Y] = G. Now we set

s(ip)=a -(-1,,)+G (2) where G is a map which realizes i/i. Then s(tp), clearly, realizes 2 it = W. Moreover, s(w) does not depend on the choice of qi. For this we first remark that we have the central extension of groups

Ext(A,FB)>G .Hom(A,B) (3) with i(a) = 0 + a. Moreover, we have

+ a = +1(a) (4) where + on the left is the action and where + on the right is the group structure of G. Now consider a second realizationi/r of 4. Then clearly iP = i/i + a for appropriate a and we get with -1 = - lyy:

(-1)4i=(41+a)-1)(+/r+a) (5) _(+/,+a)-[(-1)0 +(-1)*a] (6) +ia-ia-1)r/r=+/i-(-1)aG. (7) In (7) we use (4) and the fact that (-1)* =id: iB-alB for-1: (8) In (6) we use the linear distributivity law for the action +. 420 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

Next we show that s in (2) is actually a functor. Let cpo E Hom(A0, A) with cPo = 2 qio. Then we get:

s(W)°s(Wo) = (+/1- -(-1)k)

(9)

For a = 2 0*0 we have the realization

a=(iG-(-1)i/r)c1i.

Moreover, we get for the second summand in (9)

-((-1)r -¢r)i/ro= -(-1)a. (10)

Therefore (9) and (10) show

s(cp)°s(cpo)=a-(-1)a=s(wcP0) since 2 a = cpcpo.

(12.7.2) TheoremThe composite functor

Ab[1/2] - M2[1/2] 7r4 Ab is naturally isomorphic to the functor which carries A to 1'T(A) ® L3(A, 1) and (p: A -> B to FT(cp) ® L3(cp,1). The composite functor

(M3)°P X Ab[1/2] 1 X-so(M3)'P X M2[1/2] Ab isnaturallyisomorphictothe functor whichcarries (M(B, 3), A)to Ext(B, L3(A,1)) ®1'T#(A, B) and ( fir, cp)toExt(r/i, L3(cp,1)) ®rT0(pi, cp). The isomorphisms are both compatible with the corresponding operators A and

We do not describe the isomorphisms explicitly since we derive Theorem 12.7.2 from the following fact.

(12.7.3) LemmaH'(Ab[1/21, Hom(FT, L3(-,1))) = 0. Proof Let 5:Ab[1/2] - Hom(TT, L3(-,1)) be a derivation. We have to 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 421 show that 3 is an inner derivation. For cp: A --> B we have 2B cp = cp2A where 2A: A -A is multiplication by 2. Hence we get 3(2B (p) = 5((p2A) where 5(2BCp) = cp*5(2B) + (2B)* 5(0 = cp*3(2B) + 83(w) 6(cp2A) = cp* 3(2A) + (2A)*3(0 =cp*5(2A)+45(cp). Hence we have 45((p) = cp* 3(2A) - cp*6(2B) or equivalently 8(cp) = cP*(1/4)5(2A) - cp*(1/4)3(2B) and hence cp is an inner derivation.

Proof of Theorem 12.7.2Since A is uniquely 2-divisible also F(A) is uniquely 2-divisible and hencerr4 M(A, 2) = ir4 M(A, 2). By (11.1.17) the natural sequence L3(A, 1) >-> ir4s(A) -. FT(A) is split for each A. We choose a splitting sA and obtain a derivation 8 as in the proof of Lemma 12.7.3 by

S(cA) = or4s((p)sA -sBI T((p). This derivation, by Lemma 12.7.3, is an inner derivation. Hence we can alter the splitting in such a way that we obtain a natural splitting. In a similar way one can prove the second part of Theorem 12.7.2. Now let I'Ab[1/2] be the full subcategory of I'Ab consisting of all quadratic functions 71: A -> B with A E Ab[1/2]. Moreover let types2'[1/2] be the full homotopy category of all K67,2) with 71 E TAb[1/2]. Then there is a functor

(12.7.4) s: I'Ab[1/2] - types2' [1/2] which is a splitting of the functor k2 in (7.1.8). We obtain the functor s by the equivalence K2: typesZ = I'M2 in Theorem 7.2.7 and by the functor s in Proposition 12.7.1, that is, s in (12.7.4) carries to K2'{cp1,O,scpo}. Using the splitting s and the linear extension (7.1.8) one obtains the equivalence of categories

(12.7.5) r: types2' [1/2] -=a rAb[1/2] x E 422 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA where the right-hand side is the canonical split extension for the natural system E with E(n, n') = Ext(A, B'); see Definition 1.1.9(d). A morphism in FAb[1/2] X E is given by a tuple X = (gyp,, po, t ; ): n -->1' where (cp,, qPo): n -> n'is a morphism in rAb and where C E Ext(A, B'). Now let K = rAb[1/2] x E and let

k: K K - Ab be the following structure on K, see (12.5.1). The functor k is the projection. The functor r, carries n: A - B E Ob(K) to the direct sum

(a) r,(1) = r, (n) ® rT(A).

Moreover r, carries a morphism X = (gyp,, (po,): n - n' to the induced map r,(x): r2(,7)ED rT(A)- r2(,1')®rT(A') given by the coordinates r; (p rT(Vo) and rT(A) 10 rPpB'®A'-Lr2(ii').

Compare (11.4.8). Next the functor F3 in (12.7.6) carries the pair of objects (H,,7) to the direct sum

(b) r3(H, n) = Ext(H, 1722(77)) ED rT,(H, A).

Now r3 carries a morphism (i/i, X) with +/!: H' - H E Ab to the induced map ry *X * where

* = Ext(ii, r2 (n)) ® rT#(+G, A) and where

X*: Ext(D, r2(,7)) ® FT*(D, A) - Ext(D, F2(77')) ® FT*(D, A') has the coordinates Ext(D, F2(90' 9,)), FTO(D, go), and

rT,(D, A) h'-> [do, dA ®A] Ext(D, B' ®A) Ext(D, r2n').

Here we set ((po)* = Ext(D, q(1 ®cpo)) where q(1 ® qpo) is defined as in (12.7.6). Compare (11.4.9). There is an obvious (0, A)-short exact sequence for F3. 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA 423

(12.7.7) Lemma Using the equivalence r in (12.7.5) the structure (12.7.6) is good in the sense of (12.5.4).

Proof This is a consequence of Theorem 12.7.2 and Theorem 11.4.7. See (11.3.7) and (11.3.8).

The lemma shows that we can apply the classification theorem 12.5.7:

(12.7.8) TheoremLet AZ-systems be defined by the structure on K = TAb[1/2] x E in (12.7.5) above. Then there is a detecting functor from the homotopy category of simply connected 5-dimensional CW-spaces X for which H2 X is uniquely 2-divisible to the category A 3 -Systems(K).

12.8 The case H2X free abelian

We here consider simply connected 5-dimensional homotopy types X for which H2X is a free abelian group. Let Ab(free) be the full subcategory of Ab consisting of free abelian groups and let M2(free) be the full homotopy category of Moore spaces M(A, 2) with A E Ab(free). Then one has the equivalence of categories

(12.8.1) Ab(free) = M2(free) defined by the homology functor H2. We now consider the functors ?r4, 1r3 on M2(free). Recall that we have the equivalence of categories G: M3 = G in Theorem 1.6.7; for abelian groups A, B the group G(A, B) is the set of morphisms A -. B in G.

(12.8.2) TheoremThe composite functor Ab(free) = M2(free) -` > Ab is naturally isomorphic to the functor F22; see Lemma 11.1.7. Moreover the composite functor

G°P x Ab(free) = (M3)°PX M2(free) is naturally isomorphic to the functor which carries (B, A) to the direct sum Ext(B, L3(A,1)) ® G(B, T(A)) and cp) to Ext(r/r, G(t/r, r(q) )). The isomorphism is compatible with 0 and A.

Proof Since FT(A) = Owe obtain 7r4 M(A, 2) = r2(A) by (11.1.9). Moreover the generalized Hopf map

7lA [M(I'(A),3), M(A,2)] is natural in A, therefore we obtain the result on 1r3 by (11.2.3). 424 12 SIMPLY CONNECTED 5-DIMENSIONAL POLYHEDRA

Let TAb(free) be the full subcategory of FAb consisting of all quadratic functions ri: A --> B with A E Ab(free). Let typesz(free) be the full homotopy category of all K(q, 2) with 77 E FAb(free). Then the functor k2 in (7.1.8) yields the equivalence of categories

(12.8.3) T: types;(free) -> FAb(free).

Now let K = rAb(free) and let k: K -rAb (12.8.4) 174: K --> Ab r3:G°PxK- Ab be the following structure on K; see (12.5.1). The functor k is the inclusion and I'4 is defined by r; in Definition 11.3.3, that is r4(71) = r2 ('q). Moreover r3 carries the pair of objects (H, r7) with q: A --> B to the abelian group r3(H, ri) defined by the push-out diagram

Ext(D, r(A) ® 71/2)°G(D, r(A)) Ext(D,q)I push 1 Ext(D, r267)) » r3(D, 17)

A morphism (ii, cp) withr l r E G and c p = (apt, W2) E FAb induces4, ',p * = Ext(i', r2(Qp)) ® G(i/i, r(cpa)).

(12.8.5) Lemma Using the equivalence T in (12.8.3) the structure (12.8.4) is good in the sense of (12.5.4).

Proof We apply Theorem 11.3.4 and (11.3.8) and Theorem 12.8.2 above.0

The lemma yields the following application of the classification theorem 12.5.7.

(12.8.6) TheoremLet AZ-systems be defined by the structure on K = rAb(free) in (12.8.4) above. Then there is a detecting functor from the homotopy category of simply connected 5-dimensional CW-spaces X for which H2 X is free abelian to the category A Z-Systems(K). Appendix A PRIMARY HOMOTOPY OPERATIONS AND HOMOTOPY GROUPS OF MAPPING CONES

A CW-complex is built by attaching cells. The attaching maps yield elements in homotopy groups, it (X"), where X" is a CW-complex of dimension< n. Here X" is also formed by attaching cells, say X" = Y Ug em, so that we have to consider homotopy groups of the form 7r,(YUg em). The computation of such homotopy groups in terms of g is therefore one of the obstacles to analysing the interior structure of a homotopy type. Very little is known about such groups. For example groups of the form 7r"(Sk Ug em), given by elements g E Trm_1(S") in homotopy groups of spheres, are very hard to compute. The main results of this appendix yield a method to compute such groups via an Eg Hg Pg sequence. This sequence generalizes the classical EHP-sequence. In fact, James introduced the EHP-sequence to study homotopy groups it"(EA) of a suspension. We introduce the EgHgPg sequence to study relative homotopy groups it (Cg, B) of mapping cones Cg where g: A -+ B. If B = * is a point then Cg = Y.A is the suspension, and, in this case, the Eg Hg Pg sequence coincides with the EHP-sequence. We describe the operators Eg, Pg, Hg explicitly in terms of primary homotopy operations, in particular, the operator Pg is induced by a complicated sum of Whitehead products. This very explicit expression for the operator Pg is of great importance in applications. The proof of the Eg Hg P8 sequence uses a new combinatorial model Ng of the fibre of the inclusion ig: B c Cg. The sophisticated proof relies on the geometric bar-construction and quasi-fibration techniques. We use the model Ng also for the proof of the surprising homotopy equivalence EP,a=(YEA)xflCg/(*) XflCg.

Here 11Cg is the loop space of the mapping cone Cg and P;is the fibre of the inclusion ig: B c Cg with g: A -> B, A = IA'; see Theorems A.8.2 and A.8.13. In the first three sections we introduce and study primary homotopy operations which satisfy intricate distributivity laws. With respect to addition +: [Y.A,U] x [EA,U] - [EA,U] we have for example the left distributivity law (obtained in Baues [CC]) (X+y)°f=X°f+y°f- E C"(X,y)y"f n22 426 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES where f E [IX, 1A]. More generally we deal in Section A.9 with the action +: [Cg, U] x [IA, U] * [Cg, U] and we describe for f e [ I X, Cg ] an expansion for (u + y) a f with u e [Cg, U ]. The distributivity laws of primary homotopy operations have a long pro- gression in the literature starting with the work of Whitehead, Hilton, Barratt, James, etc. In Baues [CC] we explored the connection of such distributivity laws with classical commutator calculus of group theory. Recently, in his thesis, Dreckman [DH] described for the first time the complete list of distributivity laws of primary homotopy operations.

A.1 Whitehead products

We recall some basic definitions and facts. Throughout let a space be a pointed space of the homotopy type of a CW-complex. Maps and homotopies are base-point preserving. The set of homotopy classes X - Y is denoted by [ X, Y ];it contains the trivial class 0: X -> * E Y. The groups of homotopy classes

1r,,'(X)=[E"A, X],n>_ 1, are equipped with various operations. We here study the Whitehead product, the cup products, the Hopf construction, and the partial suspension. For the product A X B of spaces we have the cofibre sequence AvB' AxB 7 AAB. Here A V B= A x{*} u{*} X B is the one-point union and A AB=A X B/A V B is the smash product. The n-fold products will be denoted by All =A x x A and A ^ " =AA ... A A. The cofibre sequence of A v B c A X B yields a short exact sequence of groups

(A.1.1) 0->[I(AAB),Z] Y. (AxB),Z]-[1A,Z]x[1,B,Z]-0. Let PI, P2 be the projections of YE(A X B) onto EA,Y.B. The Whitehead product

[ , ]: [IA,Z] x [IB,Z] - [E(A AB),Z] (1) is defined by the commutator (I7r)'([a,B])= -p*a-p*0+pia+pzP APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 427 for a E [I A, Z], 9 E [IB, Z]. Compare Baues [CC]. Let i1, i2 be the inclusions of YEA and 1B into Y.A V EB respectively. Then

wA,B=[i1,i2]E7r1 (2) is the Whitehead product map for which we have [a, (3 ] = wA B(a, 16)- We say that an element a E V B) is trivial on B if the retraction r2 = (0,1): A V B -> B carries a to 0, that is (r2) *(a) = 0. For example the Whitehead product map WA B is trivial on lB. Let

V B)2 = kernel (r2) * : V B) -> 77nxB be the subgroup of all elements trivial on B. We have for n >_ 1 the partial suspension

E: V B)2 1(.A V B)2 (3) defined by E = j -1(7r V 1) * a-':

I(7rv 1).

J V B)2

Here 7r: (CA, A) - (MA, *) is the quotient map and the isomorphisms d and j are obtained from the exact homotopy sequences of pairs of spaces; see (2.1.3). The Whitehead product map is compatible with the partial suspension, that is

(A.1.2) PropositionEwA, B = wEA,B. This is proved in (3.1.11) of Baues [OT]. From the definition of the Whitehead product we obtain the following commutator rule in the group [UX1 x xX'),Y]. For a ={a1 <

(A.1.3) - a(lpa) -13(lpb) + a(1pa) + /3(lpb) = [a,16 }Ta,b(IP0U b) where Ta,b: I AX.Vb-Y.(AX,,) A(AXb) 428 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES is defined by T,,, b(t, Xaub)=(t, x Xb) with xa = (xa1,... ,X,,)- Clearly, if for iE a n b, Xiis a co-H-space then Ta,b = 0 since the reduced diagonal DA: A -A A A is null-homotopic if A is a co-H-space. For any three elements a, b, c of a group G we have the Witt-Hall identities

(A.1.4) (a, b c) _ (a, c) (a, b) ((a, b), c) ((a, b), ca) ((c, a), b`) ((b, c), ab) = 1 where (x,y)=x-ly-'xy and z,c = (z, x) the first equation in (A.1.4) yields the equation (A.1.5)((x, y), zz) _ ((x, y), (z, x)) ((x, y), z) ((x, y), z), (x, z). For elements ai E [I Xi, Z] we define the element corresponding to (A.1.5) by

(A.1.6)W'( a1, a2, a3) _ [[al, a2],[a3, al]]T1231 + [[al, a2], a3]

+ [[[a1, a2], a31,[a1, a3]]T12313 in the group [IX1 A X2 A X3, Z]. Here the shuffle T = Tni....,,, : XI A ... A Xk -- Xn' A ... for n,,...,n,E(1,...,k) maps the tuple (x1,...,xk) to x,,,). Clearly, T1231 = 0 if X1 is a co-H-space and T12313= 0 if X, or X3 is a co-H-space. The suspension IT is also denoted by T We now derive from (A.1.4), (A.1.5), and (A.1.3) the following Jacobi identities for Whitehead products.

(A.1.7) PropositionThe general Jacobi identity for Whitehead products is W(al, a2, a3) + W(a3, al, a2)T312 + W(a2, a3, al)T231 = 0-

(A.1.8) CorollaryIf Xl, X2, X3 are co-H-spaces the Jacobi identity is

[[a1, a2], a3] + [[a3, a1], a2IT312 + [[a2, a3], a1 ]T231 = 0.

(A.1.9) CorollaryIf X2, X3 are co-H-spaces the Jacobi identity is al, a2], a3] + [[a3, al], a2]T312 + [[a2, a3], al]T231

[[ al, a2 ], [ al, a3 ]]T1213

Moreover, we need the following properties of the Whitehead product map WA. B. For pairs of spaces (A c X, B c Y) we define the product (X,A)x(Y,B)=(XxY,XxY) (A.1.10) XXY=XxBUAxY. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 429

We consider the mapping

(A.1.11) h: UAxB)- CAxCB=CAxBUAxCB which is defined by adding the following homotopies: Hl: I x A X B (t, a, b) (t, b)) H3:IxAxB--'CAXB, (t,a,b)-((t,a),b) H4:IxAXB-AXCB, (t,a,b)-(*,(t,b)). Clearly, h=-H,-H2+H3+H4 is defined on VA X B) and is null-homotopic on VA V B). Therefore h defines, up to homotopy, a unique map h such that

(A.1.12) YEA AB CAxCB homotopy commutes. The map h is a homotopy equivalence which we call the join construction. One easily checks that the quotient map jo: (CA, A) (CA/A, *) = (IA, *) yields the composition (A.1.13) WA,e=(joxjo)h:IAAB-->IAVlB which is the Whitehead product map. We say that YcX is a principal cofibration with attaching map g: A --> Y if there exists a homotopy equivalence X = C. under Y where Cg = YUg CA is the mapping cone of g. Since there is a homotopy equivalence C(CA X CB) = CA X CB under CA X CB and since CA X CB 'oxfo EA x l,B I CAxCBI IAvIB is a push-out diagram one readily gets

(A.1.14) Lemma EA V lB C EA x lB is a principal cofibration with attach- ingMap WA,B. 430 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

We can iterate this process. Consider the product of cones

(CX1, X2) x ... X (CXn, Xn) = (Pn, PR ). Then there is a homotopy equivalence

h In-1X1 (A.1.15) A...AXn=PR which we obtain inductively by (Pn-PR-X(CXn,X.)

(CPR-1,Pn_1) X(CXn,Xn) = I(PRAX,,). h We call (A.1.15) the iterated join construction; see Baues [IJ]. Now the product of suspensions (EX1,*)x... x(IXn,*)_(Tn,7, ) yields the pair (7,, 7,,) where 7,,is called the `fat wedge'. As in Lemma A.1.14 one can show that the inclusion 7,,° c Tn is a principal cofibration with the attaching map

(A.1.16) wn : I n -1 X1 A ... A Xn --> Pn "- 7n h to This map is called the nth-order Whitehead product map. Here jois the restriction of the n-fold product jo x ... X jo: CX1 x ... x CXn --> IXl x ... x IXn. Thus the mapping cone of w,, is homotopy equivalent to Tn. Further operations we need are the (geometric) cup products. The exterior cup products are pairings (A.1.17) #,#:[IX,MAIx[MY,Y,B]-->[IXAY,Y.AAB] defined by the composites a#,8:Y.XAY °L>FAAY A- YEA AB a#/3: Y.XAY XAPY.XAB °nB EAAB where a A Y = a A l . and where A A 6 is the map 'A A 0, up to the shuffle of the suspension coordinate. The interior cup products are defined by composing with the reduced diagonal Q: A -+ X A X, (A.1.18) u, u : [.X, A] X [Y.X, B] --1, [IX, 1A A B] APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 431 where a U 13= (a#13X0) and similarly a U /3 = (a#J3)(Y).Ifa, /3in (A.1.17) are co-H-maps we have a#/3 = a#B and for a' e [YEA, Z] and /3' E [IB, Z] we get in this case

(A.1.19) [ a'a, /3'/3 ] = [ a', /3'](a#/3 ). If a and 0 are not co-H-maps we have to use, instead of (A.1.19), the Barcus-Barratt formula. We now use the exact sequence (A.1.1) for the definition of the Hopf construction Hf. Let f: A X B --' Z be given and let (a,/3):AVBcAXB-f ).Z be the restriction of the map f. Then by (A.1.1) there is a unique homotopy class (A.1.20) Hf E [ EA A B, F,Z ] for which (ITr)*(Hf)= -pl(1a)-pZ(13)+(IfLet AL: 01, x fL --> O L be the loop addition map for the loop space f1 L and let

(A.1.21) HAL E [IfIL A f1L,1SZL] be its Hopf construction. Moreover, let

(A.1.22) R = RL : Ill L -" L be the evaluation map with R(t, o-) = v(t). We have the following connection with the Whitehead product.

(A.1.23) PropositionLet i1, i2 be the inclusions of 1K and L into (Y.K) V L respectively. Then [[i1,i2RL],i2RLl = [iI,i2RL]°(KAHAL) in [Y.KAIlL A OL,1KvL].

This result can be proved in the same way as (3.1.22) in Baues [OT].

A.2 The James-Hopf invariants

For a connected space B let 1(B) be the infinite reduced product of James. The underlying set of 1(B) is the free monoid generated by B - { * }. The topology is obtained by the quotient map U B" --->1(B) nZ0 432 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES mapping a tuple (b1, ... , bn) of the n-fold product B" = B X ... x B to the word (7rb1) (irbn) where r(*) denotes the empty word and ?rb = b for b E B - { * }. The subspace Jn(B) = nr(Bn) is the n -fold reduced product of B and B =J1(B) generates the monoid J(B). Let i: B -> SIB be the adjoint of the identity of I B. James [RP] has shown that the map

(A.2.1) g=gB: J(B)--'fl1B with g(b) = i(b) and g(x o b) = g(x) + i(b) for x E J(B), b e B - { * }is a homotopy equivalence. The map g induces the isomorphism of groups

(A.2.2) [Y.A, IB] = [A, f11B] = [A, J(B)], a -* a. There are mappings

gr: J(B) -->J(B ^ r) (A.2.3) gr(b1 ... bn) = fT ba A ... A ba. a where the product is taken in the lexicographical order from the left over all subsets a = (a1 < < a,} of {1, ... , n}. The James-Hopf invariants are the functions

(A.2.4) yr: [IA,jB]-*[IA,5:B^r] induced by gr, that is y,(a) = (g,) where we use the operator a H a in (A.2.2). Clearly y1 is the identity.

Remark The map g, in (A.2.3) can be defined with respect to any 'admissible ordering' of the set of finite subsets of N = (1, 2,}, see (I.1.8) and (11.2.3) in Baues [CC]. The lexicographical ordering from the left (resp. from the right) is an example of an admissible ordering.

Let g: 1J(B) -> Y.B be the adjoint of g in (A.2.1). Then the composite IMY.B) = IJ(B)_L>B is homotopic to the evaluation map Ra8 in (A.1.22). Moreover, 8r=yr(g):IJ(B) -- Y..BAr

is the adjoint of g, in (A.2.3) and

(A.2.5) JB: ISZIB=Y.J(B) V EB^r r2I APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 433 is a homotopy equivalence. Here G = E,Z 1 j,8,is the limit of the finite subsums and Jr is the inclusion of EB A r into the wedge. We will use the following formulas.

(A.2.6) PropositionFor a composite

fg:IX -- with A and L connected and X finite dimensional, we have in [12X,12 L" n] the formula

E(yn(fg))=E( E rn(f)oYr(g)) r2-1 where

r.(f)= E Yi(f)#...#Yi,(f).

'I.... ,i,z 1

For the proof of this formula see Boardman and Steer [HI]. Proposition A.2.6 corresponds to:

(A.2.7) PropositionLet K be a co-H-space and let B and L be finite dimensional and connected spaces. For a map 13: EB --> IL and for the homotopy equivalences in (A.2.5) the diagram

EKA0$ EKA ,f1EB EKA SIEL =IKAJ, .KAJL V IKABAr V IKALAnI r21 r($) nz1 homotopy commutes where

r(0)1YKAB^'= E

indenoted the inclusion of EK A L A " into the wedge. 434 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

(A.2.8) PropositionLet A be a co-H-space and let B be a connected space. Then for the reduced diagonal 0 on 111B the diagram V LAAB^" (B LAAfIB nZI EA A i EAAfY.B111AfIB V 4AABAmAB^n An(e- n,m2I JJ) homotopy commutes. Here A is defined by

AiIA A B ^" = E i#a,#b(Y.A A Ta b) aUb=n where we sum over all pairs (a, b) of non-empty subset a, b c n = { 1, ... , n) with a U b = n. The shuffle map Ta,b is described in (A.1.3) and im.n is the inclusion of >AAB^mAB^"

(A.2.9) PropositionLet K be a co-H-space and let L be a connected finite dimensional space. Then the Hopf construction HI AIL in (A.1.21) for the loop addition map on 0EL makes the following diagram homotopy commute

Y.K A 111L A fIlL P Y,K A fUL KA HAIL =IK^(JL#JL) =IKnJL V YEKALArALAS V IKAL^` r,sZ 1 t21

Here p is the folding map given by the identity L ^ r A L AS = L A ` for r + s =t.

We leave the proofs of Propositions A.2.8 and A.2.9 as exercises.

A.3 The fibre of the retraction A V B --i- B and the Hilton-Milnor theorem

Let B + = { *) u B be the disjoint union of the base point * with the space B. We have (A.3.1) A>4B=AAB+=AxB/* xB. If A is a co-H-space with comultiplication µ then also A >4 B is a co-H-space with comultiplication iL >4B: AxB-+(AVA)> B=A>4BvAXB. This yields the homotopy equivalence (A.3.2) ii: A A B --- A v A A B APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 435 by µ = (pr V 7r)( µ >4 B). Here pr and Tr are the projections of A X B onto A and onto A A B respectively. Let q: P,-A be the principal fibration with classifying map f: A - B; the space P, is also called the homotopy theoretic fibre off or simply the fibre off. We have P,= {(a,o-)EAxB'If(a)_o-(0), * =o(1)) where B' is the function space of all maps I = [0, 1] -- B. For the retraction r2: A V B - B we have the following result.

(A.33) PropositionThere is a canonical homotopy equivalence 7r:P,2-.Ax11B.

Proof By definition of P,2 we have P,2=AxfIBu * xWBC(AvB)xWB where WB = (c E B' I o (1) = * } is the contractible path object. Since f)B C WB is a cofibration the quotient map ir: P,2 -- P2/WB =A >4 f2B is a homotopy equivalence. Compare also B. Gray's proof of the Hilton-Milnor theorem in Gray [NH]. 0

Let g0:A>4fB=P,,-AvB be given by a homotopy inverse of 7r in Proposition A.3.3. We thus have the fibre sequence A>4flB -A VB- B which yields a short exact sequence of homotopy groups

0 --+ ira (A A4 fIB) it (A V B) t-2-'' iro (B) - 0 where X is a suspension. This shows that q0 induces the isomorphism (a) (q0) * : -rro (A >4 flB) _= iro (A V B)2 = kernel(r2) We therefore get the following corollary of Proposition A.3.3.

(b) CorollaryLet A be (a - 1)-connected and let Y be a subcompler of the CW-complex X with X" C Y C X where X'is the d-skeleton of X. Then the homomorphism 1rk(AVY)2- 7rk(AvX)2 436 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES induced by Y c X is surjective for k < a + d - 1 and is an isomorphism for k4(f2YUedU ... ) =(A )1f1Y)Ue°+dU... where we may assume that A is a CW-complex with A° -' _ * . Now the result follows from the cellular approximation theorem by use of (a) above.

Using the isomorphism in (a) the partial suspension E, defined in (A.1.1X3), can be expressed by the usual suspension 7. as follows.

(A.3.4) PropositionLet X be a suspension, then rr0 (A>4f1B)Qo- 7ro (A VB)z IF IE 1r'(Y.AVB)2 a1(1A>4fB)v commutes. Proof We prove the result only in case A is a suspension. In this case commutativity of the diagram is a consequence of Proposition A.3.5 below and of Proposition A.1.2.

(a) CorollaryLet A be (a - 1)-connected. Then E: lrk(A V B)z -> 'Tk+1(EA V B)2 is surjective fork < 2a - 1 and is an isomorphism fork < 2a - 1.

This follows from Proposition A.3.4 by using the Freudenthal suspension theorem. We remark that Corollary A.3.4(a) is also a consequence of Theo- rem A.3.10 below. For the fibre of the retraction r2: (EA) V B -+ B we get:

(A.3.5) PropositionThe diagram

µ Pre 1AXf1B1AVEAAf1B

(.A) V B (ii.lil.i2 RB1) APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 437 homotopy commutes. Here i,, i2 are the inclusions of L,A and B into LA V B, and RB is the evaluation map in (A.1.22), [ , ) is the Whitehead product and µ is defined in (A.3.2). Proof Consider the diagram CIIB -_-->WB

1i 1P R Y.SZB B where j is the pinch map and p(o,) = Q(0). There is a mapping r which extends the inclusion II B C WB over the cone and for which the diagram homotopy commutes relative to 0 B. With r we obtain the homotopy commutative diagram lur P, =LAXIIBUO,BWB F= LAxIIBuCIIB

yI 1A >a 1 B jizPrUi.J

90 LA V B f LA V EIIB id v R1, Proposition A.3.5 now follows from the general fact that (LA) AB

LA XBU8 CB L.AVLAAB LAVLB homotopy commutes. Here j: CB -EB is the pinch map and p is the quotient map.

In addition to Proposition A.3.5 we have the following result for the fibre of the retraction r2: LA V L.B -+ Y.B.

(A.3.6) PropositionLet A be a co-H-space and let B be connected. Then the diagram

P,,=1AVLAAfILB V SAAB 438 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES homotopy commutes. Here 1r = µir as in Proposition A.3.5 and J = 1 V (A A JB) as in (A.2.5). The mapping W is given by the r -fold Whitehead products

WIYA A B-= [... [i1,i2],...,i2]. Forr=O we set WIEA=i1.

Proof We have to show W -(A AJB) = [i1,i2R$], but this is a consequence of 11.3.4 in Baues [CC]. Proposition A.3.6 is the basic step in the proof of the Hilton-Milnor theorem. To state the Hilton-Milnor theorem precisely, we need a certain amount of formal algebra (we follow the excellent presentation of Boardman and Steer [HI]). Let B = B1 V V Bk be a one-point union of co-H-spaces. Take abstract symbols z1, z2,....zk and let L be the free Lie algebra (over 7L) generated by the letters z1,..., Zk. Let F be the free non-associative algebraic object generated by z1, ... , Zk with one binary operation [ , I. F is the set of `brackets' or of `formal commutators' in the letters z1,..., zk. There is an obvious map F - L which we suppress from the notation. The weight wt(a) of an element a E F is the number of factors in it. By induction on weight we define for each c E F the space

B, ifc =z, ACB = (AQB)A(AbB) ifc=[a,b] and the iterated Whitehead product ww E [I Ac B, 1B] by

wc=the class of the inclusion EB, c YB ifc = Zr [wG,wb] ifc=[a,b].

For a family of spaces (Pa) let na P. be the direct limit of the finite subproducts. It is well known that the free Lie algebra L is a free abelian group and that there exists a subset Q of F which yields a base of the free abelian group L; such a set Q is called a set of basic commutators.

(A.3.7) Theorem(Hilton-Milnor): Let Q be a set of basic commutators and give Q any total ordering. Then the map fl f1wc: fl fl! A` B - 111B, CEQ CEQ

defined by using the multiplication in 01B in the order indicated by Q, is a homotopy equivalence. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 439

The following recipe for the construction of a set of basic commutators, Q, is available. We define and order Q inductively. The elements of weight 1 in Q are the elements z1, ... , Zk with z 1 < ... < zk. Now suppose that all elements of weight < w in Q are defined and ordered. Then an element in Q of weight w > 1 is a bracket [a, b] where wt(a) + wt(b) = w, a < b, and if b = [c, d] then c S a. The elements of weight w are then ordered arbitrarily among them- selves and are greater than any element of less weight. Proof of Theorem A.3.7 We indicate the proof for the wedge A V B of two connected co-H-spaces (B1 =A, B2 = B). Since r2: A V B -+ B is the retraction we obtain by Proposition A.3.6 the isomorphism V IAAB^r rZ0

1I (!z), + W. (*) ir,,(IA V IB) Here the group V SB') with A' = v (A A B ^ r I r,-> 1} and B'= A has again a splitting as in (*). This way we obtain inductively the proposition of the Hilton-Milnor theorem. Since we assume A and B to be connected the connectivity of the fibres is raised by the inductive steps. Such considerations are also valid if A and B are not co-H-spaces. Let g: A - B be a map. The fibre of the retraction A V B - B is the first approximation of the fibre Pig of the principal cofibration ig: B ti Cg. To see this we consider the commutative diagram of unbroken arrows:

A >4 OB f-rr Pri -a Pio t0) PiR

` 19 I I (A.3.8) AvB = AvB -4(g,1) B li, 1r2 ho O B Z CA V B (erg 1)Cg

Here all columns are fibre sequences. Clearly, the map a, induced by r2, is a homotopy equivalence. Thus for the map To, induced by (7rg,1), we obtain T = To a -11r1 by homotopy inverses of a and ir. The subdiagram O is a push-out diagram which defines the mapping cone Cg; the map io is given by the inclusion A c CA. For the inclusion V c: W we have the natural isomorphism of relative homotopy groups irj (W,V)Ira (Pi) 440 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES where P; is the fibre of is V C W. This isomorphism carries the homotopy class of a pair map F: (CX, X) - (W, V) to the homotopy class of the adjoint map F: X -> P, with F(x) = (F(x), ox), ox(t) = F(t, x). Now diagram (A.3.8) shows that the diagram of homotopy groups

(A.3.9)

irX(CAvB,AvB) i(Cg,B) (7"0.` commutes. Here we assume that X is a suspension and a CW-complex. A map is V- W is n-connected if the fibre P; is (n - 1)-connected or equivalently if i * : ir;V -> zr;W is an isomorphism for i < n and an epimorphism for i = n. If is V - W is n-connected we know that i * : fro (V) -''moo(W) is an isomorphism if dim X < n and an epimorphism for dim X 5 n. This is easily seen by the cellular approximation theorem since we may assume that V is a subcomplex of the CW-complex W and that V contains the n-skeleton of W.

(A.3.10) TheoremLet g: A -+ B be a mapping where A is (a-1)-connected. Then T:AA12B-'P,, is (2a - 1)-connected, or equivalently (7rg,1)* in (A.3.9) is an isomorphism for dim X < 2 a - 1 and an epimorphism for dim X S 2 a - 1.

Using (A.3.9) this result is a special case of the general suspension theorem (V.7.6) Baues [AH], see also (3.4.7) Baues [OT]. A different proof of Theorem A.3.10 can be obtained by Corollary A.6.3 below. We point out that the maps q and q0 in (A.3.8) can be replaced by the corresponding maps in Propositions A.3.5 and A.3.6 provided A and B are suspensions. Moreover, the map r in (A.3.8) has the following property.

(A3.11) Lemma The diagram AAfIBpA it r1

pi" -- f) 5:A A APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 441 homotopy commutes. Here i is the adjoint of the identity on YA and A is induced by the pair map jg: (Cg, B) --* (Y. A, *).

A.4 The loop space of a mapping cone The loop space l B is the subspace of B' consisting of all paths o : I - B with u(0) = o-0) = *. The loop space is an H-space by the addition of paths; this addition, however, is not associative. Therefore, we also use the following loop space of Moore which is a topological monoid.

(A.4.1) DefinitionLet QB+= {r E QB j r >_ 0). The Moore loop space SIB is the subspace of BR* x IR+ consisting of all pairs (o,, k,) where o,: if- B is a map with r(0) = o (t) for t >_ ko. The space SIB is a topological monoid by addition

(Q,ko)+(T,k,)_(Q+T,ko+k,) with (o+TXt)= r(t) for t_0. The inclusion is SIB C SIB, r - (0, 1) is an H-map and a homotopy equivalence.

We now consider the loop space 11C5 of a mapping cone Cg where g: YA -+ B is defined on a suspension YA. In this case we have the following commutative diagram

A -+ CA

(A.4.2) SIB flCg n n n`8. SIB SICg

Here g is the adjoint of g and 4rg is the adjoint of

agT : Y. CA -+ CY, A -* Cg where T interchanges the Coordinate and the s-coordinate. The map irg is given by the definition of a mapping cone, see (A.3.8), and ig: B C Cg is the inclusion. 442 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

We derive from (A.4.2) the following push-out diagram in the category of topological monoids: !(t° J(A) J(CA)

1(g). push 1 (A.4.3) fZB Mg

Here (g) and (erg) are the extensions of g and irgin (A.4.2) which are homomorphisms of topological monoids; J(A) is the infinite reduced product of James, see Section A.2. For the push-out Mg we have the unique map m which is a homomorphism of monoids and for which diagram (A.4.3) commutes.

(A.4.4) TheoremAssume that the space A is connected and that B is 1- connected. Then the map m in (A.4.3) is a homotopy equivalence of spaces. We call Mg -+ flCg the model of 12 Cg.

ProofUsing the work of Adams and Hilton [CA] and Lemaire [AC] we see that m induces an isomorphism in homology and hence is a homotopy equivalence since Mg and fIC5 are connected. The theorem can also be considered as a very special case of the model construction in Baues [GL]. Theorem A.4.4 is the basic step for the CW-models of loop spaces of Husseini [TC], [CR] and Toda [CS].

We obtain the topology of Mg by the surjective quotient map

f1BX(CAXf1B)n (A.4.5) ir: U -Mg n20 with where b;E S1B and a; E CA. The monoid multiplication on Mg is represented by

(b0, a,, b,,..., an, bn + b0, a,,..., dn bn). Clearly, the spaces

(A.4.6) Mn=or(SZBX(CAXf1B)n) give us the filtration SIB=M0cM, c cM c CMg APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 443 of Mg. We obtain M from by the push-out diagram of spaces in (A.4.7) below. For the pairs of spaces (X, A), (Y, B) we define the product as in (A.1.10). Moreover, let ((CA) n' (CA)on) _ (CA, A) X X (CA, A) be the n-fold product as in (A.1.15) and let

(fZB)"+'-* T: (CA)° X (d1B) x (CA X flB)n be the canonical permutation of coordinates. The mapping zr in (A.4.5) yields the commutative diagram

T((CA)*" X (S B)"+') C U1 B) X (CA x fl B)" (A.4.7) 1- C M

This diagram is a push-out diagram of spaces. For example M, is given by the push-out diagram flBxAXhB'fBxCAXf2B (A.4.8) l- I f2 B M1 with ir(bo, a, b) = bo + (ga) + b, for a e A and with g as in (A.4.2). The pinch map j: Cg -> Cg/B =12 A has the followingproperty:

(A.4.9) PropositionThe diagram

Mg m fl f C9

J(Y.A) f SZyZ A homotopy commutes. Here g is the equivalence in (A.2.1) and j is defined on Mg by j(ao,b,,...,b,,,

where jo: CA - CA/A = IA is the quotient map. 444 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES A.5 The fibre of a principal cofibration Let g: IA - B be a mapping as in Section A.4 where 0B and A are connected. For the principal cofibrationig: B N Cg we have the fibre sequence

-> 11B flCg P. -B-Cg; compare (A.3.8). The map i is the inclusion of the fibre q' (*) = fl Cg. In Section A.4 we constructed the model

Mg -=a 11Cg for the loop space 11Cg. We now consider the fibre Pi, and we define a model a Ng for the space P,as follows.

(A.5.1) DefinitionLet - be the equivalence relation on Mg which is generated by x - i(b) x for b E fiB, x E Mg. We define Ng by the quotient space Ng = Mg/ - . Let v: Mg-Mg/'=Ng be the quotient map.

(A.5.2) TheoremThere is a canonical homotopy equivalence n such that the diagram Cg -. P;` =Im =In

Mg Ng homotopy commutes.

We prove this theorem, which is a basic result of this chapter, in (A.5.10) below. The filtration of Mg in (A.4.6) induces the following filtration on P. Let N,, = v(M,,) be the image of the subspace M,, of Mg in (A.4.6). Then clearly we have the filtration No= * cN,cN2c ... cN,,c ... cNg on Ng. From (A.4.8) and Definition A.5.1 we derive

(A.5.3) N, = CA x fIB/A x f2B = (IA) x flB. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 445

(A.5.4) PropositionThe mapping T in (A.3.8) makes the diagram (IA) X flB = N, C Ng =T(-1)bi =In (IA) )4 SZBT P's homotopy commutative.

We prove this result in (A.5.11) below.

(A.5.5) TheoremThe inclusion

N.- I CN. is a principal cofibration with attaching map

wn: In-1AAn X (flan)-An where (1 Bn = f I B x ... x fl B is the n fold product.

Proof The mapping vir with v in Theorem A.5.2 and IT in (A.4.7) gives us the push-out diagram of spaces (compare (A.4.7)):

(CA)°" X (11B") c (CA x (1B)"/(* x f1B)n (1) I Nn-1 C Nn

Here we use the fact that Pa(* x (IB)n = * by Definition A.5.1. We omit the obvious permutation T in (1); see (A.4.7). Now iin (1) is a closed cofibration into a contractible space. Therefore i is equivalent to the inclusion into the cone. Moreover, we have the homotopy equivalence

n (CA)onIn-1Ann (2) which is the iterated join construction in (A.1.15). Now the mapping wn is given by (on = vir(hn X (S Bn)).

The results above are also available for the trivial map g = 0: IA - * = B. In this case Ng is essentially the reduced product space J(IA) in (A.2.1). The 446 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES reduced product filtration J"(Y.A) corresponds to the filtration of Ng above. The inclusion (A.5.6) J"_ ,(IA) CJ"(IA) is a principal cofibration with attaching map

(A.5.7) W": y" 1Ann

Here W" factors through the higher-order Whitehead product map

W": In-'A _T° 'rJ"1(.A) where (Tn, 7 (I A, *)"; see (A.1.16). The map -7ris the projection in (A.2.1). In particular

W2 =I'MAI 11A I:IA AA -1A is the Whitehead product for the identity 1YA of Y.A. Theorem A.5.5 shows that Ng has an iterated mapping cone structure which generalizes the one of J(IA). We now compare Ng and J(IA) by using the quotient map jg: (Cg, B) (1Z A,*). We consider the homotopy commutative diagram

Ng n P; 8 (A.5.8) Ij lA J(IA)noZEEA where A is induced by the pinch map jg as in Lemma A.3.11 and where no = (flT)gfA = (cl(-1))gEA

is given by the homotopy equivalence 9Y Ain (A.2.1). The map T: 12 A -> 12 A with T(t1, t2, a) = (t2, t 1, a) is homotopic to - 1 where 1 is the identity on 2 A. We define j in (A.5.8) by

j(b1, al,..., bn, an) = (j0a1)..... (Joan) where jo: CA - la is the pinch map; see Proposition A.4.9. We derive from Proposition A.4.9 that (A.5.8) actually homotopy commutes. Moreover, by the proof of Theorem A.5.5 we obtain:

(A.5.9) PropositionThe map j in (A.5.8) is filtration preserving and the pair map j: (Nn,Nn-1) -+(Jn,Jn-1) APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 447 with Jn =Jn(EA) is a canonical map between principal cofibrations, that is: the attaching maps can be chosen such that the diagram In -1AAnA1Bn--'APr I W" Iw Nn- Jn-1 commutes and there are homotopy equivalences for which the diagram

C., CWI

1s -1 Nn -' Jn homotopy commutes relative to Nn_ 1.Here j is the map J= j U C(pr). Recall that the map Cr: CU --> CV denotes the cone on r: U -p V with (Cr)(t, u) = (t, ru) for t E 1, u E U.

(A.5.10) Proof of Theorem A.5.2For a path-connected topological monoid M let BM: A°P -* Top (1) be the geometric bar construction; see (1.1.5) in Baues [GL]. The realization of this simplicial space, IBMI, is a classifying space for M. As in (1.1.6) in Baues [GL] we have the quasi-fibration M --->IEMI -->IBMI (2) where I EM Iis contractible. A space B gives us the homotopy equivalence

IBSIBI = B. (3) For the inclusion is f2B c Mg of monoids we consider the pull-back

i'IE,yal c IEMRI

19 1 (4) B=IBfBI C IBMgI i Clearly, we have a homotopy equivalence n1 for which

Mg -m-> S1Cg

(5) i*IEM I -, pi R nl x 448 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES homotopy commutes. We now define a map n2 for which

(6)

Mg/' = Ng ? i* I EMgI is commutative and we show that n2 is a homotopy equivalence. First we observe that i* I EMSI is the realization of the following simplicial space F, F: 0°P - Top

n F(t(n)) _ (f1B)xMg F(do) =pr1: (SIB)" xMg - (c1B)n-1 xMg

F(di) = µi: (SIB)" x Mg -- (SZB)"-1 x Mg (7) fori = 1,2,...,n F(si) =Ji+1: (SIB)n1 xMg - (1B)- xMg fori = 0,1,...,n - 1.

Here di are the face operators and si are the degeneracy operators in the simplicial category°P; see §1 in Baues [GL]. Moreover

AL'i(X1,...I XI) = (xi,...,xi'Xi+I,...,xn) pri(x...... xn)= (xl,...,xi-1,Xi+l,...,x") li(x1,...,xn)= (x1..... Xi-1, *,xi,...,xn).

We have by definition in the proof of 1.1.6 in Baues [GL] the canonical equivalence

I FI = i* I EMSI (8) of realizations. The projection q in (4) is the realization of the natural transformation

q: F -'BSIB (9) with

rt rt q =prn+ 1: F(O(n)) = (SIB) X Mg --> (SIB). (10) APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 449

We now define n2 in (6). Let

n2: 1F1 -> Ng (11) be induced by the projections

W x MB)n XMg ->Mg-v->Ng.

We have to check that n2 is well defined. In fact, this is clear by definition of µn above and by Definition A.5.1. Moreover, it is clear that n2 makes (6) commute since j in (6) is given by

Mg = Do x F(O(0)) c IFI.

We now claim that

IEnBI C IFI Ng (12) is a quasi-fibration with fibreI En B I. Since I En B I is contractible we conclude that n2is a homotopy equivalence. Thus by (5) and (6) we have the proposition in Theorem A.5.2. For the proof of (12) we consider the restrictions:

IFIn-1 cIFIn c IFI

IPA-1 lPn (13) c11N,, c Nglnz N.-1 with IFIn = nZ'(N,). By definition of F in (7) and of n2 in (11) we see that

IFIn = IEiiBI. (14)

Now we assume that pn is a quasi-fibration with fibreI En B I. In (1) of the proof for Theorem A.5.5 we obtained a push-out diagram for the inclusion Nn_ 1 CN,, which gives us the push-out diagram

n I1B)n (CA)°n x (nB)c (CA x

Vir I- (15)

N,, C N.

It is clear from the definition of F, that if we pull back IFI, over vir, we have

n (vir)' IFIn = IEn8I x (CA x (IB). (16) 450 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

Therefore we obtain a push-out diagram

I EfBI X (CA)°" x (flB)" C IEnBI X (CA X (B)"

1 I IFI"- 1 c If 1. lying over the push-out (15). The push-out property of quasi-fibrations (see Hardie [QA]) gives us the result that p" in (13) is a quasi-fibration. Thus also nZ is a quasi-fibration (see V. Puppe [RH]).

(A.5.11) Proof of Proposition A.5.4Since the mappings in Proposition A.5.4 are equivariant with respect to the action from the right of f1B we obtain the result by proving that the following diagram homotopy commutes:

Now T I )A corresponds to the map Y .A - WCI A which is adjoint to the identity of CIA. However, nI EA corresponds to the map s: IA - WCIA given by s(T,a)(r) = (0,1 - 2r + 27-r, a) E=- CIA for 0 B = Y .B' is a map between suspensions we gave a different proof of Theorem A.5.2 in Baues [RP]. This proof relies on a construction of Gray [HM] which is only available if B is a suspension. All results in Baues [RP] are special cases of the results in this chapter.

A.6 EHP sequences

For a map g: IA ---> B we have the isomorphism (A.6.1) (7rg,1)*: 7r"(CIAVB,Y.AVB)-7r"(Cg,B) APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 451 between relative homotopy groups. In this section we embed (1rg,1) * into a long exact sequence which generalizes the EHP sequences of James. This shall give us a fundamental tool for the computation of the groups 1r (Cg, B). Ganea [GH] and Gray [HM] also studied the homotopy groups of a mapping cone. The improvement here is the fact that the exact sequences below are available in a considerably better range of dimensions. The map - 1: Y,A -> )A induces an automorphism of 1r (I A) which we denote by (-1) *.

(A.6.2) TheoremLet A be connected and let B be 1-connected. Then we have a commutative diagram

(n6. )) a.+i(Cg,B) -Y' NI)a+ 1r,(CMA V B, I.A V B) -a

I -lid 1).(j8). 1. vB) 1).(18).

(y2A) - A,1.A) ' v E Y in which the rows are long exact sequences. Here jg is the pinch map and j: Ng -> J I A is defined in (A.5.8); r, : I A V B -,, I A is the retraction.

Clearly, if B = * is a point all vertical arrows in the diagram of Theorem A.6.2 are isomorphisms. The bottom sequence in the diagram is the classical suspension sequence of James [ST] which is given by the use of the equivalence 9Y,,: flY.I.A =JY.A in (A.2.1). The operator y is the composite

Y 2 A) = A) (g4 -)7r (JEA) I A).

The operator d is the usual boundary operator and I is the suspension. We now define the operator y by

B) = N)

Moreover, the operator a is the composite

see (A.3.8). For N, = IA A 0 B the map -1: N, -+ N, is - 17A A 12 B. 452 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

Proof of Theorem A.6.2 We consider the map of pairs

j: (Ng,N,) -> (JXA,LA) which induces a map of the corresponding long homotopy sequences. Now the proposition is a consequence of Theorem A.3.10, Proposition A.5.4, and Theorem A.5.2.

This shows that a result as in Theorem A.6.2 is also available for the hornotopy functorinx instead ofIr,,; compare (I1.7.8) in Baues [AH]. If 1A is (a - 1)-connected we derive easily from the iterated mapping cone structure of JY.A in (A.5.6) that (JI.A, Y.A) is (2a-1)-connected. Therefore the exactness of the James sequence gives us the classical suspension theorem of Freudenthal. More generally, (Ng, N,) is also (2a - 1)- connected, therefore we obtain from the exact row in Theorem A.6.2 the following special case of Theorem A.3.10.

(A.6.3) CorollaryLet EA be (a - 1)-connected and let B be simply connected. Then Or.,1), in (A.6.1) is epimorphic for n < 2a and an isomorphism for n <2a.

We can apply this corollary to the principal cofibration N, c N2 in Theo- rem A.5.5. The attaching map for N2 is

(A.6.4) C02: (YA AA) >4(SZB)2 - (YA) >c fiB = N, and thus we obtain (compare Theorem A.6.2):

(A.6.5) CorollaryLet EA be (a - 1)-connected and let B be simply connected. Then

Or.,, 1). d-': IT,-,(XA AA >a (flB)2 V EA >4 f1B)2 -> N,) is epimorphic for n < 4a - 2 and is isomorphic for n < 4a - 2.

Clearly from the iterated cone structure of Ng we see that

(A.6.6) N,) --> N,)

is epimorphic for n:!5; 3a-1 and is isomorphic for n < 3a - 1 in case Y .A is (a - 1)-connected. Thus in the appropriate range of dimensions we can replace the group 7r (Ng, N,) in Theorem A.6.2 by the groups in Corollary A.6.5. This leads to the following corollary of Theorem A.6.2. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 453

(A.6.7) TheoremLet YEA be (a - 1)-connected and let B be 1-connected and let g: IA - B. For n < 3a - 1 we have the following commutative diagram with exact columns:

7rn-I(Y.AX DB) (- 1). pr. IEg 1y 7rn(Cg, B) A)

IH8 1IH 7ri-2(X.A AA >a 1ZB2) P-' 7rn-2(T.A AA)

IPS IP

7rn-2(>A >a SZB) 7Tn-2(Y.A) (- I )* pr. IEg 11

The map pr is the projection and we set

Pg=(-1)*(w2)*, P=[1YA,1EA1*

Moreover, for n = 3a - 1 the following commutative diagram extends the diagram above such that the columns remain exact:

773a-2(Y.AAA >i OB2 V IA)2 7r3a-2(I A AA V IA)2 (prv1)1). I(-1).('Z.id. 1([1%AI1fA' 11A). IT3a-2(IA > fB) 7r3a-2(EA) (- 1). pr.

AddendumFor the operator Eg the diagram

7rn_ 1(Y.A >U SIB) µ 7rrt_ 1(IA v I,A A f1B)

I --g (g, [g, RBI). 7rn(Cg,B) 7T,-,(B) commutes, where µ is defined in (A.3.2) and where [g,RBI is the Whitehead product of g and of the evaluation map RB: I fl B - B. If B is a suspension we can replace [g, RB) by use of Proposition A.3.6. 454 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

Proof The operator Eg is defined by

iTn-,(Y.A >4 f1B) 7Tn 1(P,) = 7rn(CIA V B, 1A V B) E(arn).

I (irg,1). 7rn(Cg, B) or equivalently by Eg: 7rn-,(1 A >i flB) - . 7rn- t(P,x) _ B) where T and Ira are defined in (A.3.8). The operator Hg is defined for n+1<3a-1by 7rn+,(Cg, B) _ ?fn(Pi,) n- iTn(Ng) xj Hg 7rn(Ng,NI) 7rn- I(IA AA >4 1ZB2) 7rn(N2, Nl) where we use Corollary A.6.5 and (A.6.6). The operator H is similarly given by

A) = 7rn(flY.A) (gIA). IH 7rn(JY.A, Y.A) 7T,,-,(Y.A AA) 7rn(J2GA, 1A) where we use the equivalence in (A.2.1). If B is a point all horizontal arrows of the diagram in Theorem A.6.7 are isomorphisms and one easily checks that in this case the diagram of Theorem A.6.7 commutes. The proposition of Theorem A.6.7 is a consequence of Proposition A.5.9, Theorem A.6.2, and Proposition A.3.5. We have

7r3a_2(Y.A AA >1 )B2 V IA)2 = 7r3a-2(Y.A AA >4 11B2 V EA >a ,fiB)2 since 7r3a-2(1A AA AA A fIB)=0. 0

(A.6.8) PropositionFor the operator H in Theorem A.6.7 and for the James-Hopf invariant 72 the following diagram commutes:

7r (1(IA)) - 7rr: (T.(F,An AYEA)) H =I(-0. 7rn-2(111Y-A AA) 7rn(Y.3A AA) APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 455

Here v(t1, t2, t3, a, b) = (t1, t2, a, t3, b) for t, E I and a, b EA. Since EA is (a - 1)-connected and since n < 3a - 1 the operator II is an isomorphism.

By the isomorphism (q0)* in Proposition A.3.4 we derive from Theorem A.6.7 the following exact sequence in which the operator Pg has a simple description.

(A.6.9) TheoremLet g: IA - B and let IA be (a - 1)-connected and B be (b - 1)-connected, b >- 2. Moreover, let A be a co-H-space. For n < min(3a- 1, 2 a + b - 1) we have the exact sequence 7r,-1(Y.AVB)2- Eg Hg P" Eg Here Hg is defined in Theorem A.6.7. Eg is given by

Eg = Eg(go.)-' = (7rg,1)*d-', and the operator Pg is induced by a Whitehead product:

Ph = (q0)* Pg = [i1, i1 - i2g] * where i 1 and i2 arethe inclusions of Y .A and B respectively into 1A v B. If n = 3a - 1 < 2a + b - 1 the exact sequence has a prolongation 7r3a-2(LA AA V ).A)2- 7r3a-2(LA V B), with Pg = ([i1, i1 - i2g], -i1)*. For the operator Eg the diagram 7rii-1(EA VB)2 C7ri-1(Y.A vB) IEa I(g. 1).

7r,,(Cg,B) a- 7r, -1(B) commutes. Here d isthe boundary operator. As in Theorem A.6.7 the (EE, P;, Hg)-sequence forms a commutative diagram with the (E, H, P)- sequence.

Proof By the assumption on n we know 7ri_2(Y.A AA) = 7ri-2(1A AA>4 fB2). Moreover, we use the isomorphism (q0)* in Proposition A.3.4. We have to check that (qo)*Pg=[i1,i1 -i,g]*. 456 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

Here we use the formula for 0g2 in Theorem A.7.4(a) in the next section. Now (q0)*Pg is induced by

(q0(-1)(02)I£AA A =(11, PI ,[2RB])w 2 = (i1, [i),i2RB])([a,a] -NIA A g)) = [i1,i1I - [i1,i2RB](EA Ag) = [i1,i1] - [i1,i2g] _ [i1,i1-i2g].

Here we use RB(Ig) =g in (A.1.19), and we use the bilinearity of the Whitehead product.

A.7 The operator Pg

The operator Pg is induced by the mapping (-1)02. This mapping can be described as follows. By the homotopy equivalence µ in (A.3.2) we obtain the homotopy commutative diagram

(A.7.1) )A AA X(flBxfIB) YEAAIB Y.AXfIB

Y.AAA A1AAA A(fIBXfIB) µ -1µ

LAAAA(S°vf2Bvf2BV(OB)n2) 8 i, L,AVL,AAf1B where 02 is defined in Theorem A.5.5. The homotopy equivalence µ is given by

µ0: XX1 XX2) =Y.X1 V Y.X2 V Y.X1 AX2 (A.7.2) µ0 =i1(Y-p1) +i2(y'p2) +i3("r) which we easily derive from the exact sequence (A.1.1). The homotopy commutative diagram (A.7.1) defines up to homotopy the map 0g. This map has the components

(A.7.3) Wg =(W12, 0g23, 0g24, 0g234) APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 457 given by the restrictions of wg to the factors of the V -product. We denote by a: IA c IA v LA A fIBandb: I.A A f1B c EA v IA A SIB the inclusions and by HB=HAB: the Hopf construction on the loop addition map; see (A.1.21). Moreover, for n1,...,n,E(ml <

(A.7.4) TheoremLet A be a co-H-space. Then for a map g: Y. A -> B and its adjoint g: A -* fIB the components of wg are given by the following formulas:

(a) for o2 e[1X1 AX2, IA VIA AfT1B] we have w912=[a,a]-b(IAAt);

(b) for 1.823E[IX1AX2AX3,IAVIAASIB]we have 1.123 = [b, a]T g 132 b(A A H)(IAB A SIB Ag-)Tt32 ;

(c) for 1.824E [IX1 AX2 AX4, IA VIA A SIB] we have

(0824 = [b, a]T142 + [a, b] + [b, b]T1424 - b(A A HB)(IA A g A SIB);

(d) for (08234 E [1X1 AX2 AX3 AX4, EA V IA A SIB] we have

1.8234 = [b(AA HB ), a ]T1342 + [b(A A HB ), b ] T13424

+[b, bIT1324 -b(A AHB)(A Af1B AHB)(IA AfIB AgAfIB)T1324.

Thus, all components of 1.g are expressed solely in terms of the Whitehead product, the Hopf construction HB, and the adjoint g. Clearly, by (A.7.1), we can replace the operator Pg in Theorem A.6.7 by the operator (wg),, with wg described in (A.7.1) and Theorem A.7.4. 458 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

Proof of Theorem A. 7.4For a product X = X, x ... x X of spaces and for E { 1, ... , n} we denote a map

T: (Y.X)/- - (Y-(X; X ... Xx1 ))/= (1) by T = T,... j, if [t,x; ,...,xi.]= for t E I, x; E X;. Here - and = are equivalence relations and [x]-is the equivalence class of x in (M/=. Now let X =A xA x lB x 0B. We derive from the definition of w2 in Theorem A.5.5 and from (A.1.12) that the following diagram homotopy commutes

FX --- U = EX, V T.X, A (X2 X X3 X X4) V IX2 V EX2 AX4

1T,?,. I- EX, AX-, > (X3 xX4) TA(- Ow, IAVIAASlB (2)

Here we have m = (a,b(Y.A Amg),a,b) with mg:X,XX3XX4-+flB mg(a,b,,b2)=b, +g(a)+b2. (3)

The map is the sum

°P1 +P134 +P2 +P24 -P1234 -P1 -"P24 -P2 (4) where p; is T followed by the inclusion into U. In particularP 134is the composite A(X2XX3XX4)CU. T,34 Let j12: "I x X2) -> F,X be the inclusion. Then we have

G12 =P1 +P2 -P12 -P1 -P2 (5) with P12 =P134j12.The commutator rule (A.1.3) shows that Sj12t = (-P1, -P2) -P12 = (P1,P2) -P12- Therefore we have

Ti (wg2)=m*Sj12=T1*2([a,a]-b(IAAt)) (6) and thus (a) is proven. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 459

We now compute Pg = Wg i1 fAA A,, Ywith Y= flB x SIB; see (A.7.1). As in (A.7.2) we have the equivalence

µo: Y'(X2 X Y) = Y-Uo U0 =X2VYVX2AY. This gives us µo:U-_"U'=1X1 VY.XIAU0V1X2V1XZAX4. We consider

Tz34Pg= -mG12+mf=m*(-6112+0 (7) Let mo: U' - IA v IA A SIB be defined by moµo=m. (8) Then we derive from (7), (5), and (4)

T234Pg= (MO)* CO (9) where o= -NLo6J12+/l0 -(PI +P2 -P12 -P1 -P2) +P1 +P134 +P2 +P24

-(P12 +P134 +P1234) -"P1 -P24 -P2 (10) Now the commutator rule (A.1.3) shows

6o = (P]341P2) + (P134,P24) -P1234 + (PI'P24) where ( , ) denotes the commutator. Clearly, all these commutators correspond to Whitehead products lying in the group [I X1 A X2 A Y, U']. Using the definition of the Hopf construction H. = HAB in (A.1.20) we are able to compute from (11) the term (mo)*(60). By (9) this yields the formulas (b), (c), and (d) in Theorem A.7.4.

(A.7.5) CorollaryLet g: IA -B be a map where A is a connected co-H-space and where B is simply connected. For the mapping (dg in (A.7.3) the composition (g,[g, RB])wg is null-homotopic.

Proof The following diagram homotopy commutes EAxSIB=-=F,AxfB µ IAVIAASIB

i r I (g,[g. RB1) n 4 Ng Pi, B 460 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

Compare Proposition A.3.5, (A.3.8), and Proposition A.5.4. Now the corollary follows from Theorems A.7.4 and A.5.5.

The relation in Corollary A.7.5 is not a `new' homotopy relation since

(A.7.6) TheoremThe formula (g,[g,RB])*wg=0 can be proved by use of the Jacobi identity (Corollary A.1.9), the relation in Proposition A. 1.23, and by (A. 1.19).

Proof Since g = RB(1g) we see from (A.1.19) that

(g, [g, RB ]) * &,g2 = [g, g] - [g, RB ](EA Ag) is trivial. Moreover, we derive from Proposition A.1.23 and (A.1.19) the equations

(g,[g,RB])*wg23 =[[g, RBI,g]T132 -[g, RB](A AHB)(Y.A A flB Ag)T132

= [g, [g, RB]] - [[g, RBI, RBI(Y.A A 11B Ag)TI32

= 0.

Similarly, we derive from Corollary A.1.9, Proposition A.1.23, and A.1.19 the remaining two equations for (c) and (d) in Theorem A.7.4. For (d) we apply Corollary A.1.9 with a3 = [g, RBI, a1 = RB, a2 =g.

We now consider the special case where g in (A.7.1) is a map between suspensions. For a map g: I.A -p lB (B connected) we derive (A.7.1) the mapping Wg for which

IA AAAf1Afl 109 YEAAfl

= 1 T1324

(A.7.7) IA AAA A fl AAJB =I(A AJB)# (A AJB) IA AB* AA AB*wIA AB* APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 461 homotopy commutes. Here we set

S2=S°vf lB, B*= V B^", Bn0=S°. n20

The homotopy equivalence A A JB is described in Proposition A.3.6. Let

i": for n0 be the inclusion.

(A.7.8) TheoremLet A be a co-H-space. Then for n, m 0 the map Wg in (A.7.7) satisfies the formula

WWIEAAB""'AAAB""= F,[im+#a, #b]va,b aub=n lr+m+n(A AB Ar A')'r(g) AB A "). r2I

The first sum is taken over all pairs (a, b) of subsets a, b c n = (1,... ,n} with a U b = h. 7he shuffle map

va b: Y.AABAm AAABAn-IA AB ^(m+ra)AA ABA#b is defined by

V.,b(t, u, X, v, y) = (t, u, X A ya,V,Yb) for t E I, u, v E A, x E B " M, y E B ^ ";compare (A.1.3). The James-Hopf invariants yr(g) are defined in Section A.2.

If B is a co-H-space Va, b is trivial if a f b is non-empty. Theorem A.7.8 is a corollary of Theorem A.7.4 by use of (A.1.19) and Propositions A.2.8 and A.2.9. We proved Theorem A.7.8 in a slightly different way in Baues [RP]. Again we can replace the operator Pg in Theorem A.6.7 by (A.7.7) and by the operator (Wg),R where Wg is given by the formula in Theorem A.7.8. This is of great value for the study of the groups lr"(Cf, B) for n:5 3a - 1. Explicitly we obtain the following result which is a special case of Theorem A.6.7.

(A.7.9) TheoremLet A be a co-H-space. Let YEA be (a - 1)-connected and let I B be 1-connected and let g: I A -. EB be a map. For the relative homotopy 462 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES groups of the mapping cone (Cg, 1B) wehave the following commutative diagram with exact columns, n 5 3a - 1.

fin-1(Y.A A B*) 'fin-1(1.A) IF, ly irn(Cg, EB) in(12 A)

In. lH 7rn-2(F.A A B* AA A B') 'r,,-2(Y .A AA)

(W,) I P 'rn-2(IA A B*) 1rn-2(Y.A) (( 1). pr. J E, lY 1rn-1(Cg, Y.B) 1rn-1(y 2A) (- 1). (la).

The map pr is induced by the projection B* - S°; recall that B* = S° V B V B A 2 V . The map Wg is defined in Theorem A.7.8. Moreover, for n = 3a - 1 the following commutative diagram extends the diagram above such that the columns remain exact:

1r3.-2(LA AB* AA AB* V IA)2 4 AA V 1A) (prvl)1).` lr3a-2(Y 1(W8.-i0). I" 11,1).

IT3a-2(IA AB*) (- 1). pr, ir3a-2(L'A)

Here i° is induced by S° c: B*.

AddendumFor the operator Eg the diagram

1rn-1(Y.A A B*) _a lE W 7r,(Cg,F.B) -a 1rn_1(1B) is commutative where

is the i-fold Whitehead product: clearly, for i = 0 we set WEA =g. Compare Proposition A.3.6 and the addendum of Theorem A.6.7. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 463

Proof of Theorem A.79 We define Eg by the composition (see Theorem A.6.7):

n B=) = >a fIB) ir,,(Cg, B) Ea where the isomorphism is induced by µ in (A.7.1) and by A A JB in (A.7.7). We define the operator Hg by the composition (see Theorem A.6.7): $B)Hiri-2(Y.A AA >4 f2(Y.B)2) __ iri-2(Y.A A B= AA A B=). Here the isomorphism is induced by µ,µ in (A.7.1) and by

(A AJB)#(A AJB)T1324 in (A.7.7).

A.8 The difference map V

The difference operator V is of importance for the computation of the left distributivity law for maps between mapping cones; see also (11.2.8) Baues [AH]. Let g: A --> B be a map between connected spaces. Then the difference operator V is induced by a map V0, namely, there is a commutative diagram

Trx(Cg, B) ir; (:A V Cg)2 v' 'II iro (P,) ) -rroY(ft(Y,A v Cg))

The map Vo is constructed as follows. The inclusions i,: EA c IA v Cg and i2: Cg c 1A V Cg yield the map 12+i1:Cg ->Y.AV Cg which we defined by the cooperation on the mapping cone Cg. We consider the diagram f2(IA A flCg) V/ (A.8.1) PigvfZ(IA V Cg) Inr2 f2Cg 464 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES with V0(b,o-)_ -i2o+(i2+i1)o for (b,o-)EP, cBXWCg,o(0)=ig(b). The element -i2 o-+ (i2 + i1)o is given by addition of paths in EA V Cg. It is clear that this element is actually a loop in Y.A V Cg. We call V. the difference map for the mapping cone Cg. It is easy to verify that Vo actually induces the difference operator V as described above. Since in diagram (A.8.1) the composition ((1r2)Vo is null-homotopic there is by (A.3.1) up to homotopy a unique map V which liftsV0, thatis (flgo)V = V0. Let

V: Y.P,, -> Y. A A f)Cg be the adjoint of V. We shall prove:

(A.8.2) TheoremIf A= YEA' is a suspension and if A' and (B are connected then V is a homotopy equivalence.

This resultis proved in Theorem A.8.13. We point out that V is a homotopy equivalence for any map g: A -* B where A and B are connected.

(A.83) RemarkIf B = * is a point we have Cg =1 A and P; = f Y.A. In this case it is well known that there is a homotopy equivalence as in Theorem A.8.2, namely

I P ;=Y. fl 1A - = + V EAAr=1AVAA(V AAr) s JIA r2I `r2I // Iv(AAJIA)

Y.A V YEA A SlLA (1)

1AAf1Cg= YEAA[l A compare (A.2.5) and (A.3.2). The equivalence, however, does not coincide with the canonical map V. In fact, we have the homotopy commutative diagram

ESZY. A YEA A f11A

=1J'-A (2) V 1AAr C a V 1AAr=1 r21 r2I The vertical arrows are the homotopy equivalences which are defined by the row and the column of diagram (1) respectively and c is the map with Here cr(j1, ...,jr) is the element defined in the proof of APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 465

Theorem A.9.5 below. By (6) in the proof of Theorem A.9.5 we see that c is a homotopy equivalence.

We define the map A by the homotopy commutative diagram D(Y.A x fly) x flY --A-;I l(IA x fly) (A.8.4) =j =j 1(Ax f1Y) x lY --1(AX fly) where the homotopy equivalences are defined as in (A.2.1). The map A is given by the formula

A((a1,o )...(an,a,,),a)=(a1,a1 +a)...(a,,,o +a) where a1 E A, a;, a E fly. It is easy to see that A is a well-defined map. The map A is used in the following diagram P;8xfC5 yx1,l(IAX1lCg)xfCg IAsVi IA flCg cP;` ffl A x flCg) where µ is the operation and i is the inclusion. The map A ® Vi is defined by (x, y) H A(x, y) + (Vi)y where + is the addition of loops in ft(Y. A x f2Cg).

(A.8.5) TheoremThis diagram homotopy commutes.

Proof We consider v°x 1, P',a x lC8 MIA V C)8 X IICg

(1) 1, ®v, F; yu f2(IA V Cg) R with I(x,a)= -i2a+x+i2a. We set (I®VoiXx,a)=I(x,a)+Voi(a). Now (1) homotopy commutes as we see by the homotopies Voµ((b, a), a') = -i2(a+ a') + (i2 +i1)(a+ a') = -12a' -i2a+ (12 +11)a+ (i2 +i1)a' -i20'+V0(b,a)+i2a'+V0(*,a'). Now f1g0 in (A.8.1) induces a monomorphism (lq0)* on homotopy sets. Therefore Theorem A.8.5 follows from the following lemma and (1). 0 466 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

(A.8.6) Lemma For a space Y the diagram

f2(MA >a IIflY) x OY tngo)x I SZ(EA V Y) x flY

lA 1, 11(EA >a f1Y) 09o f2(Y,A v Y) homotopy commutes. Here I (XI o) = -'2 0 + x + i 2 o- and q0 is defined as in Proposition A.3.3.

Proof Let is A x flY- 0Y.(A >G flY) be the adjoint of the identity and let

i": (A >c f2Y)" - ffl (A >4 f)Y) i"(x...... x") =i"-'(x1,...,xn-,)+(ixn), i' =i.

It is enough to prove that for all n >: 1 we have

I((11go) x 1)(i" X 1) = (flgo)A(i" x 1). (1)

But if (1) holds for n = 1, then by definition of A we see that (1) holds for all n. In fact, we derive this from the following homotopies with xi = (a;, o;) for i = 1,...,n. (f g0)A(i" x 1)(x,,...,x",Q)

(f q0)i"((a,,o,+o),...,(an,on+o)) (2) +o) (3) = -i20'+(nqo)(aI,o,)+i20'-i2o+(lgo)(a2,o2)+ (4) -i2o+(1Zgo)i"(x,,...,xn)+i2o (5) = I(fIgo x 1)(i" x 1)(x,,..., xn, o).

Here (2) follows from the definition of A, (3) from the definition of i", and (4) is a consequence of (1) with n = 1. (5) follows from the standard homotopy o - o = 0. For n = 1 (1) is a consequence of the following lemma. 0

(A.8.7) Lemma For a space Y the diagram

A >A lY A X OY

1Z(EA >c fY)9f2(IA v Y) APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 467 homotopy commutes, where ITis the quotient map and where 10(a, o) _ -i2o+i,a+i2Q with a=iafori:AcflY.A. ProofIt is easy to see that W AxflY)+i AxflY Inµ

fl(Y.A V 1A A flY) IT

fl(EA v 10Y) - - A x flY homotopy commutes with I1(a, o) = -'2 & +'14' + i 2 &; compare Proposition A.1.2. Thus we conclude Lemma A.8.7 from Proposition A.3.5. Clearly, Ry&= o and (f1(1 v Ry))I1 = Io. For the mapping r in (A.3.8) we obtain the diagram LA x flB c EA >4 flC5 (A.8.8)

7. Pi, where the inclusion is 1 )4 flig.

Proposition(A.8.8) homotopy commutes.

ProofThis follows from Theorem A.8.5 since A x f1B - P;F x f2Cg TlAxnit Xl flB T P.lµ x homotopy commutes and since Vo(r I A) is the inclusion of A. Clearly, Vo I fl B and thus Vi Ia Bis null-homotopic. We now replace A above by the suspension IA so that g: EA -+ B. We assume that A and AB are connected. We want to give a combinatorial description VN of the mapping V, that is we want to describe explicitly a map VN for which the diagram Ng -a J(IA>4Mg) (A.8.9) P, v fl(IIA A flCg) 468 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES homotopy commutes. Here n is given in Theorem A.5.2 and m is defined by (A.2.1) and by m in Theorem A.5.2. The mapping V on P;d is defined in (A.8.1). Let v : Mg - Ng be a quotient map in Theorem A.5.2 and let

is Cg CMg be the inclusion of the mapping cone of g: A - f1B given by i U ITICA in (A.4.3). Each element in Mg is of the form

X, X" where x; E Cg, i = 1, ... , n, n1. Each element in Ng is of the form

[x1... v(x,... X"). Let .lo:C,- 5:A --I P Y.A be given by the pinch map. We define VN in (A.8.9) by the formula

(A.8.10)

This is the product in the monoid J()A >4 Mg) of the elements

(Iox1, x1+ I *xi+2' ... E EA >4 Mg with i = 1, ... , n. One can check that VN is a well-defined map.

(A.8.11) TheoremFor the mapping VN in (A.8.10) diagram (A.8.9) homotopy commutes.

It is easy_to see that (A.8.9) is homotopy commutative if we restrict to N, = Y.A A4 fIB; compare (A.8.8). Moreover, VN in (A.8.9) yields the following commutative diagram which corresponds to Theorem A.8.5:

NgXMg

lA. VN (A.8.12) 1" Mg Ng J(Y.A >Mg) VN APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 469

Here A is defined in the same way as A in (A.8.4) and we set

(A ® VNV)(X, Y) = A(X, Y) -VNVy and µ([X1...Xn],Xn+l ...Xm) _ [X1...Xm].

We deduce directly from the definition that (A.8.12) commutes.

Proof of Theorem A.8.11 We have to show that

Ng 7" J(Y.A A Mg ) =I-

Pi" Sl(1,1A >' ilCg) (1) ID9n H(12AVCg) homotopy commutes. Let 11 X be the loop group of Kan which is the topological group equivalent to the loop space 11X, X connected. (If SX is the reduced singular complex and if GSX is the semi-simplicial loop group of Kan [HT], then fl X = I GSX Iis the realization.) In all considerations of this chapter we can replace Cl or Cl by Cl. In particular, we can replace fl in (1) by Cl. Let is Y.A -+ fIVIA) be the `adjoint' of the identity on E2A and let

m:CgcMg---ACg,g: A-->flB.

Then the composite in (1), namely

(Slgo)mON: Ng f((y2 AV Cg), satisfies by Lemma A. 8.7 the formula _1 - v[XI ... XJ =X2 ... X. 1oX1.X2 ... Xn -1 - .X3 ... Xn .10X2.X3 ... Xn --1 - .Xn .IOXn-I*Xn*1oxn (2) 470 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES where we set x, ***X,_ (fli2Xin(x1)- - joxi= (fti1)ijox;. Since 1is a group we see

p[xl ...x ] = x1 ... zrt z,OXI)... x joz(3)

We have to show Van = V. We first give a characterization ofVpas follows. We consider the commutative diagram

l(12A vC) 4f)(C)

1 (4) B Cg

112 12AVCg 12+1, where V is the map induced on fibres. Since igq = 0 there exists a map V0 which lifts V. Now V in (1) is up to homotopy the unique lifting of t with

W r2)70 - 0. (5)

Clearly, the inclusion i2: Cg c 12 A V Cg is the principal cofibration with attaching map go: Y.A -p { *) c Cg. Therefore (D in (4) corresponds to

41 Mga = AY A) * ftCg

Iv (6) Ng --p Ng. =Mgo/fICg 6N where Mg0,is the free product of monoids and where Ngo is given by the action from the left of ,Cg on Mgo. We claim

VN[xI ...xJ _ {x]'(joxd'...'xn'(jox.)] (7) where z, = ma;. From (7) and (3) we derive that satisfies the characterization of Vo in (5) and thus we have proven Von = V. For the proof of (6) we remark that i/i in (6) is given by

t1Y1 ... y ] _ (P1Y1)_(P2Y1)'... (8) APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 471 for y; E )A V SZ Cg , i = 1, ... , n. Moreover, V. is, by the naturality of the construction Ng, induced by the map X:Cg, EAVlCg (9) i2m +ilj0.

Thus VN[x, x" [(Xx,) ... (Xx")]. This and (8) proves (7).

(A.8.13) TheoremThe adjoint of VN in (A.8.9) VN:YNg -VAXMg is a homotopy equivalence.

Proof From the combinatorial definition of VN in (A.8.9) we derive that

(ON),,: J(Ng) -->J(YE A AAMg)

induces an isomorphism in homology and thus is a homotopy equivalence. Since (V N). corresponds to 1 VN also V. is a homotopy equivalence.

Similarly we get for the filtrations in (A.4.6) and Theorem A.5.5 the following result.

(A.8.14) Theorem ON restricts to a homotopy equivalence

VN: i for n z 1. With Mo = 11 B we have V, _ -1 on.N,; see (A.5.3).

A.9 The left distributivity law

In (11.2.8) of Baues [CC] we obtained the following left distributivity law. Let X be finite dimensional and let Y be a co-H-space. Given f E [.X, X.Y] and x, y c- [Y.Y, U] we have the formula

(A.9.1) xf+yf=(x+y)f+ E c"(x,y)oy"(f). n22 Here ,,(f) is the James-Hopf invariant defined with respect to the lexicographical ordering from the left, see (A.2.4), and the terms c"(x, y) E [ Y.Y ^ ", U ] are given by

(A.9.2) c"(x, y) = F, [x,y]m(d)T,(d). deD 472 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

Here [x, yJ,1(d) is an iterated Whitehead product of weight n. The set Dn is computed in (1.1.13), (1.1.16) of Baues [CC]. For the map f: IX -+ 1Y we have the difference element (see Baues [AH] II112) Vf= -i2f+(i2+i1)f:'.X-*Y-YV Y which is trivial on the second YY, r2 *V f = 0. Therefore the fibre sequence in Proposition A.3.6 yields the diagram

V J,Y A n YY n nr n>_1

I w /r 1 (A.9.3) "f YEY V Y,Y

1(0,1)-r2 // Of Y. X IY

Since r2 *Vf = 0 and since r2 is a retraction there is a unique homotopy class Vf which lifts the class Vf, that is W*Vf-= Vf. We define the element

(A.9.4) Hnf=r,, VfE[Y.X,EY^"] by the retraction rn in (A.9.3). The element Hnf is one of the Hilton-Hopf invariants which, in particular, was considered by Barcus and Barratt [HC]. The advantage of definition (A.9.4) is the fact that Hnf depends only on the definition of Vf since the map W in Proposition A.3.6 is defined canonically. On the other hand, the James-Hopf invariants depend on the choice of the `admissible ordering'. In (A.2.4) we have chosen the admissible ordering given by the lexicographical ordering from the left. In fact this is a good choice since we prove:

(A.9.5) TheoremLei f E [IX, Y.Y] where X is finite dimensional and where Y is a co-H-space. If the lames-Hopf invariant, % f, is defined with respect to the lexicographical ordering from the left we have the formula -Hnf=(-1)o(y,,f).

Here -1 E [lY ^ ", MY ^ "] is given by the identity 1 of MY ^ ". The formula implies Y. Hnf = Y- ynf. The result In 'Hnf = E"' yn f is also proved in Theorem (4.18Xa) of Boardman and Steer [HI]. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 473

Proof of Theorem A.9.5 We deduce the theorem from formula (A.9.1). By (A.9.1) we know

Vf=ilf-L Cn(i2,il)Yn(f) (1) n22 where cn(i2, i1) E [Y A ", YYv IY] is given by (A.9.2) with r2 * cn(i2, i 1) r= 0. Therefore there is a unique

Y^n, V Y1'^kl cn(j1,...,jn)E{ (2) I kz1 J with W*cn(jt,..., jn)=cn(i2,i,). Here .Y^mC V 1Y^k jm:Y kal is the inclusion and cn(j1, .. ., in) is a sum of iterated Whitehead products of such inclusions. By the explicit formula for cn(i2, i1) in (A.9.2) it is possible to derive an explicit formula for cn(j1,..., in) in (2). For example we have

-c2(jl,j2) =j2, (3)

c3(j1,j2,j3) =j3 + [j2,j1] + [j2,jl]T132 (4) By (1), (2) and by the definition of Vf in (A.9.3) we see that

Of =J1f - E Cn(11, ,In)Yn(f) (5) nz2 We claim that for the retraction rn in (A.9.3) we have rn * cm = 0 for m not equal to n and rn*Cn(I1,...,id _ -Jn,n > 2. (6) This follows from the fact that the only bracket [x,Y]d.(d) with d E=- E D,, in which y appears only once is

[x2,y1,x3,...,X1]. (For this compare the explicit description of Dn in (1.1.16) in Baues [CC].) Now (5) and (6) yield the result in Theorem A.9.5. O

We now consider the partial suspension of the element Vf E [IX, JYv IY]2; compare Proposition A.3.4 and (A.1.1)(3). By Theorem A.9.5 we get 474 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

(A.9.6) CorollaryLet f E [ I X, I Y ] where X is finite dimensional and where Y is a co-H-space (it is enough to assume that Yis connected). With the inclusions y2Y,E2Yv yy, 5:Y i2 we have the formula in [12 X, I2 Y V I Y ]2 f) 2 n22 w h e r e [1" I2' ] = [ "" [i1> 12 ] "', i2 Bis an (n-1)-fold Whitehead product. Proof We have

EVf =E(WVf-) = (EW)(EOf) (1) where

E f-=I(E imHmf) (2) mZl by (A.9.3). By Proposition A.1.2 and by definition of W in Proposition A.3.6 we obtain the proposition in Corollary A.9.6 where we use the fact that I Hm f = Iym f. Compare the proof of (3.3.19) in Baues [OT].

Corollary A.9.6 is of importance for the homotopy classification of maps. From (II.13.10) in Baues [AH] we derive:

(A.9.7) TheoremLet C = Cf be the mapping cone off: IX - IY where X is finite dimensional and where Y is connected. Moreover, let w: Cf - U be a map with restriction u: I Y -+ U. Then we have the long exact sequence (k z 2):

[jk+1Y,U] W ' . .k(UcIw)

-` - [y2 Y, U] u'J- [I2 X, U] - [Cf, U] ---. [EY, U].

Here V"(u, f) is given by the formula

Vk(u, f)(a) = (EkVf)*(a, u) =a(ykf)+ E [a,un-1](Ikynf) n22 with ae[yk+1Y,U]andk>_1.WesetV=V'. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 475

The element[a, u" - ) ] = [. [a, u], --- u]]denotes an (n - 1)-fold Whitehead product. Exactness in Theorem A.9.7 implies that the image of V(u, f) is the isotropy group in w E [Cf, U] of the action [E2 X, U] + on the set [Cf, U], that is:

(A.9.8) image V(u,f)={f3E[12X,U]I{w}+/3={w}}. Therefore we have the bijection

(A.9.9) [Cf, U] = U [V X, U]/image V(u, f ). ue[£Y,U] f"u=0 The bijection is defined by choosing elements w in each orbit of the action [V X, U]+ on the set [Cf, U]. This shows that Theorem A.9.7 yields an explicit method for the enumeration of the set [Cf, U]. A first result of the type in Theorem A.9.7 was obtained by Barcus and Barratt [HC], see also Rutter [HC], however, in these papers only equation (A.9.8) and not the exact sequence in Theorem A.9.7 is discussed. For a mapping cone Cg, g: Y -. B, and for a map f: IX - C. we obtain a diagram similar to (A.9.3) as follows:

Y.Y A UCg EY V IY A flCg W_(ij,[i,,i,Rj) I H(f) (A.9.10) XX Y.YvoCg Ir2

Compare Propositions A.3.5 and A.3.6. Here Vf is again defined by f = -'2f+ (i2 + i1)f. Since r2 Vf = 0 and since r2is a retraction there is a unique homotopy class Vf which lifts Vf. We call

(A.9.11) H(f)=r2*VfE[IX,IYAflCg] the total Hopf invariant of f. If B = * is a point we can derive from H(f) all invariants H f in (A.9.3). Clearly, we have

(A.9.12) rt * Vf-=jof E [I X, Y_Y] where jo: Cg -- Cg/B = Y.Y is the pinch map. By the Hilton-Milnor theorem

(A.9.13) Y,Vf = i)(Ijog) + i2(Y.H(f )) where it and i2 are the inclusions. 476 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES

(A.9.14) ProblemIs it possible to express Vf solely in terms of jo f and H(f)? If B = * is a point this is true by the left distributivity law described above; see (6) in the proof of Theorem A.9.5. Theorem A.8.11 might be helpful.

We now consider the partial suspension of the difference element Vf E [IX, Y-Yv Cg 12. This generalizes Corollary A.9.6. By (A.9.10) we get:

(A.9.15) PropositionWith the inclusions

12 Y---12 Y V Cgt2 Cg Jt J we have the formula

(EVf) = j1E(jof) + [j1,j2RcB] ° (YHf ).

Thus EVf depends only on jo f and on the total Hopf invariant Hf.

Proof of Proposition A.9.15 We have E(Vf) = E(WVf) _ (EW)(EVf ) and therefore the result follows from (A.9.13) and Proposition A.1.2.

Clearly, we can derive from Proposition A.9.15 an expression for the operators V'(u, f) in the exact sequence (11.13.10) Baues [AH] in the same way as in Theorem A.9.7:

(A.9.16) Proposition f [IX,Cg], u: Cg --> U, iB E [E"Y, U].

V"(u, f)(13) = (E"Vf)*( /3, u)q

uRc](y."Hf) =G(I"jof)+[13 1 where Rcx: If2Cg --> Cg is the evaluation map.

A.10 Distributivity laws of order 3

We here describe all distributivity laws of order 3 in homotopy theory. They are needed in the proof of Chapter 9. We derive these distributivity laws from 2.51 in Dreckmann [DH]. Recall that a map f: X - Y is a phantom map if for all finite dimensional CW-complexes K and for all maps g: K --> X the composite g' is null-homotopic. APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES 477

(A.10.1) NotationLet. be a class of pointed spaces and consider the homotopy group [MA,B) given by pointed spaces A, B. We define the subgroup U(3) c [EA, B] to be the normal subgroup generated by all composites EA->X-B withXE.E and all phantom maps EA - B. For f, g c- [I A, B] we write f =g modulo I if (f) = {g} in the quotient group [EA, B]/U(.).

In the following theorem we consider the primary homotopy operations +: [EA, Z] X [Y.A, Z] -+ [MA, Z] (addition) o: [EA,EB] X [EB,Z] MA, Z] (composition) #,#: [EA,1.X]x[IB,IY]--i[IAnB,IXnY](exterior cup products) U, U: [IA, EX] x [IA,EY] --> [EA,EXAY1 (interior cup products) y,,: (YEA, EB] - [EA, EBAM] (James-Hopf invariant)

[ , ]: [EA,Z] x [EB,Z] -p [EA AB,Z] (Whitehead product).

Here y,, = is defined with respect to an admissible ordering < (satisfying (1) < (2), see (A.2.4)). Moreover, let 7 be the shuffle map in (A.1.6).

(A.10.2) TheoremLet A, A', B, B', C, Z bepath-connected pointed CW-spaces. Then the primary homotopy operations above satisfy the following formulas (a)-(h).

(a) Let < and be two admissible orderings and f E [EA, EB]. Then we have for n = 2,3 modulo EB^',rz4.

(b) Let u, v c- [EB, Z] and f e [YEA, EB]. Then of +vf = (u +v)f + ([u,v]T21 + [[u,v],v]T212)y2(f )

+ ([[u, vl,v]T213 + [[u, vl,vl T312 + [[u, v], u ]T231)y3(t) modulo EB Ar z 4. (c) Let u E [EB, Z], v E ['EB', Z], f E [EA, EB], and g E [IA', IB']. Then [uf,vg] = [u,vl(f#g) + [[u,v],v](f#y2(g)) + [[u,vLu]T132(y2(f)#g)

modulo EBArA(B')A',r+s>_4. 478 APPENDIX A HOMOTOPY GROUPS OF MAPPING CONES (d) Let fE[Y.A,1B],gE[Y.A',IB']. Then f#g=f#g, and fug=fug for A = A' where both equations hold modulo EB A (B') ^', r + s 4. (e) Let f E [YA, EB]. Then fUf=T11f+(T12+T21)y2(f)

f U y,(f) = (T112 + T212)y2(f) + (T123 + T213 + T312)73(f )

Y2(f) U f = (T121 + T122)y2(f) + (T123 + T132 + T231)y3(f )

where these equations hold modulo 1B A r, r >_ 4. (f) Let f, g E [ I A, I B]. Then y2(f+g) = 72(f)+ y2(g)+fUg y3(f +g) = y3(f) + y3(g) +f U y2(g) + y2(f) u g.

These equations hold modulo Y.B ^', r >- 4. (g) Let f E [1B, Y.C], g E [1A, Y.B]. Then y2(fg) = y2(f )g + (f#f)y2(g)

y3(fg) =y3(f)g + (f #y2(f))y2(g) + (y2(f)#f)y2(g) + (f#f#f)y3(g).

These equations hold modulo Y.B A', r4. (h) Let f E [IA, AB1, g E [5A', IB']. Then y2([f,g]) = (T12 - T21)(f#g)

y3([ f, g]) _ (T121 - T112 - 7,12 + T221)(f#g)

+(T123- T213 - T312 + T321)(f#y2(g)) +(T132-T312-T213+T231)(y2(f)#g)

These equations hold modulo lB A r, r >- 4.

W. Dreckmann in 2.51 of [DH] described formulas as in Theorem A.10.2 which actually hold modulo the empty class of spaces; the reduction of these formulas yields the formulas above. BIBLIOGRAPHY

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NOTATION FOR CATEGORIES

Boldface sans serif letterslike C, A, Toptypes,(m -1)-connected(m+r)-types 55, denotecategor ies. 97 spaces;"(m - 1)-connected (m +r)- Ob(C)class o f objects dimensional CW-spaces 97 C(A, B) = Homc(A, B) = Hom(A, B)set of types;"(C) 97spacesm(C) 97 morphisms spaces(m,n),181,110spaces(m,n)H195 Autc(A) group of automorphisms types(m,n) 181types(m,n)H 195 COPopposite category Ak 295 AN-" 152ak 316 FC category of factorizations 10 M" Moore spaces M(A,n) 18 Palr(C)category of pairs 224 M2[1/2] 419 M2 (free) 423 FM" 24 C x D split linear extension 11 G 27 Set*category of pointed sets 398 Moore(m,n) 191M(m,n) 192 Gr category of groups P" 22 Ab category of abelian groups Gro(E) Grothendieck construction 84 Ab[1/2] 419 Ab (free) 423 bypes(C,F) 87b(G) 105 FAbfinitely generated abelian groups 25 Bypes(C,F) 888(G) 105 fAbfinite abelian groups 202 kypes(C,E) 83k(G) 105 Cyccyclic groups 22 Kypes(C, E) 83K(G) 105 fCycfinite cyclic groups 386 PCyc elementary cyclic groups 361PCyc° S(E°,E,) 108 s(E,,,E,) 108 402 Add(Z)finitely generated free abelian groups H,,,CW-tower of categories 122 215 Hq,,, H(Q"' 164 Add(P) 215Add(l/n) 220 Chain_ = Chain chain complexes of abelian OM(Z) quadratic Z-modules 216 QM(7L/n) 221 groups 33 rAb = rAb,quadratic functions 240, 347 Chain(R) 225 rAb[1/2] 421FAb (free) 424 Topcategory of topological spaces FAb(C) 385, 394rAb,(C) 395 Top*pointed topological spaces 32 SrAb = rAb", n >- 3stable quadratic func- Top'/=homotopy category 32 tions 240 STAB' 250rM" 245 rG 247 AAb 277 CW category of CW-complexes with trivial 0-skeleton 32 A3-cohomology 275 CW/= 33 CW" 121CW2 122 A 3-Systems, A 3-systems 254 (CWZ)"+i A3-Systems, A3-systems 282 138 spaCes2simply connected CW-spaces 56 AZ-Systems(K), AZ-systems(K) 414 n-typesn-types 54 model(T) models of a theory T 398

INDEX

A 3-cohomology system 274 cohomology Ak.-polyhedron 150 of a category 14 A3-system 253 with coefficients in n: A 0 1/2 -- B A3-system 281 of a group 14 AZ-system 413 of a space with coefficients 34 action on a functor 120 cohomotopy group 149 additive category 24 of Moore space 202, 259 Adem relation 156 cohomotopy systems 163 Adem operation 273, 312 colift 270 approximation of quadratic functor 222 comultiplication 398 Atiyah-Hirzebruch spectral sequence 161-2 cone of a space 32 attaching map 5, 35, 121 connected 5 n-connected 5, 34 path connected 5 simply connected 5 Barcus-Barratt formula 336 contractible 2 Betti number 302 correspondence 9 bifunctor 11 coskeleton 149 bimodule 11, 118 cross-effect 217 biproduct 215 cup product 430, 477 Bockstein homomorphism 14 CW-complex 5, 32 bottom sphere 22 CW-space 5, 32 boundaries in a chain complex 33 CW-tower of categories 124-5 boundary invariants 64, 130 cycles 33 theorem 66 homological 69 boundary operator for 1-groups 46 decomposable 294 Bullejos-Corrasco-Cegarra cohomology 244 decomposition 294 bype 86, 87 theorem 306 extension 90 derivation 91 functor 86 derived functor 227 detecting functor 9, 99 difference homomorphism 267, 359 difference map 463 category 8 dimension 5 category of factorizations 10, 118 disk 4 cell 4 distributivity law of order 3, 476 cellular approximation theorem 33 Dold-Thorn result 51 cellular chain complex 33 duality for quadratic Z-modules 216 cellular cochain complex 150 duality map 152 cellular map 32 duality of bype and kype 91 certain exact sequence 34, 150 duality of Spanier-Whitehead 152 chain maps, classification 37 Chang complex 296 Chang types 319 Eckmann-Hilton duality 74, 116 classification theorem 100, 110, 182, 195, 197, EHP sequence 450 283, 275, 255, 416, 417 Eilenberg-Mac Lane functor 169 cochain complex 150 bifunctor 169 coextension 270 split 184 cogroup 398 Eilenberg-Mac Lane space 55 488 INDEX

elementary Hopf construction 431 Chang complexes 296 Hopf map 17, 36 Chang types 319 Hurewicz homomorphism 7, 34, 40, 51, 52, 62, Eilenberg-Mac Lane space 319 71, 101, 151 homotopy groups 361 Moore spaces 296, 361 enriched category of Moore spaces 245 indecomposable 294 equivalence of categories 9 injective bype 87 of linear extensions 12 injective kype 83 exact sequence for functors 119 infinite reduced product 431 extension 270 infinite symmetric product 51 abelian 38 interchange map 364 exterior cup product 430, 477 isomorphism type 2 exterior square 15, 222 isotropy groups in CW-tower 143 faceI faithful 9 Jacobi identity 428 fat wedge 430 James-Hopf invariant 365, 432, 477 fibre 435 join construction 429 finite type 316 formal commutators 438 free bype 88 k-invariant 57, 73, 326 free kype 82 k'-invariant 74 full 9 knot 2 kype 82 kype I -functor 15, 223 extension 90 I--groups 34, 151 functor 81 with coefficients 41 split 82 IF-sequence 34, 109, 150, 254, 282, 414 of hype 89 of kype 85 left distributivity law 471 F -torsion 333, 173 linear extension of categories 11 graph 2 split 11 Grothendieck construction 84 linear distributivity law 11 group lift 270 of automorphisms 120 link 2 of homotopy equivalences 128 loop group of Kan 52, 469 loop space 441 Hilton-Milnor theorem 438 homeomorphism type 2 homology 33 mapping cone 210, 270 with coefficients 33 for chain complexes 39 decomposition 73 for spaces 40 isomorphism 33 metastable homotopy groups 229 homotopic 2 model of a theory 398 homotopy 32 Moore functors, bifunctors 193, 194, 258 class 32 Moore loop space 441 decomposition 72 Moore space 18 equivalence 8 Moore type 191 fibre 435 group 32, 149 with coefficients 18 natural equivalence relation 8 with cyclic coefficients 392 natural system 10, 118 rel X 33 natural transformation 14 system 122, 163 nerve of a category 13 type 2, 8 n-equivalence 101 homotopy groups, generalized 400 nilization 354 INDEX 489 obstruction operator 119, 126, 131 spherical vertex 302 opposite category 397 stable CW-tower 163 stable homotopy group 257 stable k-stem 196 partial suspension 427 stable range 149 phantom map 477 Steenrod squares 156, 312 pinch map 22 operations 271 polyhedron 1 stem Pontrjagin-Steenrod element 80 k-stem of homotopy categories 295 Postnikov functor 54 suspension functor 19, 284 Postnikov tower 56, 72 symmetric square 222 Postnikov invariant 57 theorem 58 pre-additive category 24 tensor primary homotopy operation 477 algebra 335 principal cofibration 429 square 222 principal map 211 theory, single sorted 397 projective space 257, 327 Toda bracket 209 plane 402 torsion functors 173 proper, 1 129 pseudo homology 37, 53 bifunctors 175 pseudo-projective plane 18, 21 product 171 quadratic 227 tower of categories 120 quadratic tree of homotopy types 317 chain functor 226, 172 trivial on 427 construction 223 twisted map 357 form 219 type functor of Whitehead 15, 223 n-type of chain complex 60 Hom functor 219 n-type of spaces 54, 55 map, function 15, 223, 240 tensor product 218, 227 unification problem 103 torsion product 227 unitary cohomology class 40 Z-module 215 unitary invariants 78 quotient universal coefficient theorem category 8 for cohomology 37, 38 functor 9 for homotopy groups 18, 19 for pseudo-homology 37 uniquely 2-divisible 419 realizable 9, 55 realization 10 geometrical 6 weak morphisms, isomorphisms 85, 105 reduced product 432 weight of formal commutator 438 reflecting isomorphisms 9 Whitehead right exact 228 classification theorem 106 ringoid 24 exact sequence 34, 150 square 365 theorem 6 secondary boundary operator 35 Whitehead product 17, 230, 365, 426, 477 secondary cohomology operation 272 map 427, 430 semitrivial 82, 86 Witt-Hall identities 428 sequence, (E0, E1) 107 words 298 simplex 1 special, general 299 skeleton 5 cyclic 300 smash product 152 space 32 Spanier-Whitehead duality 152, 298 Yoneda functor 399 sphere 32 Yoneda lemma 399